[ { "number": 1, "url": "https://www.erdosproblems.com/1", "status": "open", "prize": "$500", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "A276661" ], "formalized": "yes", "statement": "If $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=n$ is such that the subset sums $\\sum_{a\\in S}a$ are distinct for all $S\\subseteq A$ then\\[N \\gg 2^{n}.\\]", "commentary": "Erdős called this 'perhaps my first serious problem' (in [Er98] he dates it to 1931). The powers of $2$ show that $2^n$ would be best possible here: this provides an upper bound for the minimal such $N$ of $N\\leq 2^{n-1}$. This was improved by Conway and Guy [CoGu68] (see also [Gu82]) to $N\\leq 2^{n-2}$ (for all large $n$). The best known upper bound is\\[N\\leq 0.22002\\cdot 2^n,\\]due to Bohman.\n\nThe trivial lower bound is $N \\gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdős and Moser [Er56] proved\\[ N\\geq (\\tfrac{1}{4}-o(1))\\frac{2^n}{\\sqrt{n}}.\\](In [Er85c] Erdős offered \\$100 for any improvement of the constant $1/4$ here.)\n\nA number of improvements of the constant have been given (see [St23] for a history), with the current record $\\sqrt{2/\\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound $N\\geq \\binom{n}{\\lfloor n/2\\rfloor}$.\n\nAn equivalent formulation is to ask for the maximal size of a set of integers in $[1,x]$ which is dissociated (all subset sums are distinct). If $F(x)$ is the size of such a set then this problem is equivalent to\\[F(x) <\\log_2x+O(1).\\]Conway and Guy (see [Gu82]) conjectured that $F(2^k)=k+2$ for all large $k$, but Erdős [Er80] wrote he had 'no opinion'.\n\nIn [Er73] and [ErGr80] the generalisation where $A\\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.) This generalisation seems to have first appeared in [Gr71].\n\nThis problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] \"it is a rich kitchen where such things go to the sink\". \n\nThe sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.\n\nSee also [350].\n\nThis is discussed in problem C8 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1: [Er56][Er57][Er59][Er61][Er65b][Er69][Er70b][Er70c][Er73][BeEr74,p.619][ErSp74][Er75b][Er77c][Er80,p.97][ErGr80,p.59][Er81][Er82e][Er85c][Er90][Er91][Er92b][Er95,p.165][Er97c][Er98][Va99,1.20]" }, { "number": 3, "url": "https://www.erdosproblems.com/3", "status": "open", "prize": "$5000", "tags": [ "number theory", "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A003002", "A003003", "A003004", "A003005" ], "formalized": "yes", "statement": "If $A\\subseteq \\mathbb{N}$ has $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ then must $A$ contain arbitrarily long arithmetic progressions?", "commentary": "This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like\\[r_k(N) \\ll_k \\frac{N}{(\\log N)(\\log\\log N)^2}\\]would be sufficient. \n\nEven the case $k=3$ is non-trivial, but was proved by Bloom and Sisask [BlSi20]. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka [KeMe23]. Green and Tao [GrTa17] proved $r_4(N)\\ll N/(\\log N)^{c}$ for some small constant $c>0$. Gowers [Go01] proved\\[r_k(N) \\ll \\frac{N}{(\\log\\log N)^{c_k}},\\]where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney [LSS24], who show that\\[r_k(N) \\ll \\frac{N}{\\exp((\\log\\log N)^{c_k})}\\]for some constant $c_k>0$ depending on $k$.\n\nCuriously, Erdős [Er83c] thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao [GrTa08] (see [219]).\n\nIn [Er81] Erdős makes the stronger conjecture that\\[r_k(N) \\ll_C\\frac{N}{(\\log N)^C}\\]for every $C>0$ (now known for $k=3$ due to Kelley and Meka [KeMe23]) - see [140].\n\nSee also [139] and [142].\n\nThis is discussed in problem A5 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 April 2026. View history", "references": "#3: [Er74b][Er75b][Er77c][ErGr79][Er80,p.92][Er80c][ErGr80,p.11][Er81][Er82e][Er83][Er83c][Er85c][Er90][Er97c][Va99,1.28]" }, { "number": 5, "url": "https://www.erdosproblems.com/5", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A001223" ], "formalized": "no", "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that\\[\\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C?\\]", "commentary": "Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\\log n$. This problem asks whether $S=[0,\\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:\n$\\infty\\in S$ by Westzynthius' result [We31] on large prime gaps,\n$0\\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,\nErdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,\n Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,\n Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\\subset S$,\n Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\\%$ of $[0,\\infty)$ belongs to $S$,\n Merikoski [Me20] showed that at least $1/3$ of $[0,\\infty)$ belongs to $S$, and that $S$ has bounded gaps.\nIn [Er65b], [Er85c], and [Er97c] Erdős asks whether $S$ is everywhere dense (but Weisenberg notes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\\infty]$).\n\nSee also [234].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#5: [Er55c][Er57][Er61][Er65b][Er85c][Er90][Er97c]" }, { "number": 7, "url": "https://www.erdosproblems.com/7", "status": "verifiable", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "no", "statement": "Is there a distinct covering system all of whose moduli are odd?", "commentary": "Asked by Erdős and Selfridge (sometimes also with Schinzel). They also asked whether there can be a covering system such that all the moduli are odd and squarefree. The answer to this stronger question is no, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].\n\nHough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who also prove that if an odd covering system exists then the least common multiple of its moduli must be divisible by $9$ or $15$.\n\nSelfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).\n\nFilaseta, Ford, and Konyagin [FFK00] report that Erdős, 'convinced that an odd covering does exist, offered \\$25 for a proof that no odd covering exists; Selfridge, convinced (at that point) that no odd covering exists, offered \\$300 for the first explicit example...no award was promised to someone who gave a non-constructive proof that an odd covering of the integers exists...Selfridge (private communication) has informed us that he is now increasing his award to \\$2000.'\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#7: [Er57][Er61][Er65][Er65b][Er73][ErGr80][Er82e][Er90][Er95,p.166][Er96b][Er97][Er97c][Er97e]" }, { "number": 9, "url": "https://www.erdosproblems.com/9", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis", "primes" ], "oeis": [ "A006286" ], "formalized": "yes", "statement": "Let $A$ be the set of all odd integers $\\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?", "commentary": "Crocker [Cr71] proved that there are infinitely many odd integers not of this form; his proof in fact proves there are $\\gg\\log\\log N$ such integers in $\\{1,\\ldots,N\\}$. Pan [Pa11] improved this to $\\gg_\\epsilon N^{1-\\epsilon}$ for any $\\epsilon>0$. Erdős believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.\n\nThe sequence of such numbers is A006286 in the OEIS.\n\nIn [Er80] Erdős conjectured 'with some trepidation' that for any finite set of primes $P$ all large integers $n$ can be written as $n=m+2^k+2^l$ where $m$ is a multiple of one of the primes in $P$.\n\nSee also [10], [11], and [16].\n\nThis is discussed in problem A19 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#9: [Er77c][Er80,p.96][ErGr80][Er85c][Er92c][Er95,p.167][Er97][Er97c][Er97e]" }, { "number": 10, "url": "https://www.erdosproblems.com/10", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis", "primes" ], "oeis": [ "A387053" ], "formalized": "yes", "statement": "Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2?", "commentary": "Erdős described this as 'probably unattackable'. In [ErGr80] Erdős and Graham suggest that no such $k$ exists, although in [Er80] Erdős conjectured with 'trepidation' that such a $k$ does exist.\n\nGallagher [Ga75] has shown that for any $\\epsilon>0$ there exists $k(\\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\\epsilon)$ many powers of 2 has lower density at least $1-\\epsilon$. \n\nGranville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).\n\nSee also [9], [11], and [16].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#10: [Er77c][Er80,p.96][ErGr80,p.28][Er85c][Er92c][Er95][Er97][Er97c][Er97e]" }, { "number": 11, "url": "https://www.erdosproblems.com/11", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [ "A001220", "A377587" ], "formalized": "yes", "statement": "Is every large odd integer $n$ the sum of a squarefree number and a power of 2?", "commentary": "Odlyzko has checked this up to $10^7$. Hercher [He24b] has verified this is true for all odd integers up to $2^{50}\\approx 1.12\\times 10^{15}$.\n\nGranville and Soundararajan [GrSo98] have proved that this is very related to the problem of finding Wieferich primes, which are $p$ for which $2^{p-1}\\equiv 1\\pmod{p^2}$ - for example, if every odd integer is the sum of a squarefree number and a power of $2$ then a positive proportion of primes are non-Wieferich primes.\n\nErdős often asked this under the weaker assumption that $n$ is not divisible by $4$. Erdős thought that proving this with two powers of 2 is perhaps easy, and could prove that it is true (with a single power of two) for almost all $n$.\n\nSee also [9], [10], and [16].\n\nThis is mentioned in problem A19 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 05 April 2026. View history", "references": "#11: [Er77c][Er80,p.96][ErGr80,p.28][Er85c][Er90][Er92c][Er97][Er97e][Er97f]" }, { "number": 12, "url": "https://www.erdosproblems.com/12", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0?\\]Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\frac{N}{f(N)}.\\](Their example is given by all integers in $(y_i,\\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)\n\nAn example of an $A$ with this property where\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\log N>0\\]is given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.\n\nElsholtz and Planitzer [ElPl17] have constructed such an $A$ with\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\gg \\frac{N^{1/2}}{(\\log N)^{1/2}(\\log\\log N)^2(\\log\\log\\log N)^2}.\\]Schoen [Sc01] proved that if all elements in $A$ are pairwise coprime then\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3}\\]for infinitely many $N$. Baier [Ba04] has improved this to $\\ll N^{2/3}/\\log N$.\n\nDeepMind has provided a construction that shows the answer to the second question is no (and hence the answer to the first question is yes). This construction has been simplified and improved in the comments; it is now known that there exists an $A$ with this property such that, for all large $N$,\\[\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\geq \\frac{N}{(\\log N)^{O(\\log\\log\\log N)}}.\\]It is unknown whether there exists such an $A$ with $\\sum_{n\\in A}\\frac{1}{n}=\\infty$.\n\nFor the finite version see [13].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#12: [ErSa70][Er73][Er75b][Er77c][Er80,p.113][Er92c][Er95c][Er97][Er97b][Er97e][Er98]" }, { "number": 14, "url": "https://www.erdosproblems.com/14", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "A143824" ], "formalized": "yes", "statement": "Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.Is it true that for all $\\epsilon>0$ and large $N$\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}?\\]Is it possible that\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})?\\]", "commentary": "Apparently originally considered by Erdős and Nathanson, although later Erdős attributes this to Erdős, Sárközy, and Szemerédi (but gives no reference), and claims a construction of an $A$ such that for all $\\epsilon>0$ and all large $N$\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\ll_\\epsilon N^{1/2+\\epsilon},\\]and yet there for all $\\epsilon>0$ there exist infinitely many $N$ where\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/3-\\epsilon}.\\]Erdös and Freud investigated the finite analogue in [ErFr91], proving that there exists $A\\subseteq \\{1,\\ldots,N\\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.\n\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 September 2025. View history", "references": "#14: [Er92c][Er97][Er97e]" }, { "number": 15, "url": "https://www.erdosproblems.com/15", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "Is it true that\\[\\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n}\\]converges, where $p_n$ is the sequence of primes?", "commentary": "Erdős suggested that a computer could be used to explore this, and did not see any other method to attack this.\n\nTao [Ta23] has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.\n\nIn [Er98] Erdős further conjectures that\\[\\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)}\\]converges and\\[\\sum_{n=1}^\\infty (-1)^n \\frac{1}{p_{n+1}-p_n}\\]diverges. Weisenberg notes that the existence of infinitely many bounded gaps between primes (as proved by Zhang [Zh14]) implies the latter series does not converge. Weisenberg also has an argument which shows that, assuming the Hardy-Littlewood prime $k$-tuples conjecture, the series is unbounded in at least one direction (positive or negative).\n\nErdős further conjectured that\\[\\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)(\\log\\log n)^c}\\]converges for every $c>0$, and reports that he and Nathanson can prove that this series converges absolutely for $c>2$ (and can show, conditional on 'hopeless' conjectures about the primes, that this sum does not converge absolutely for $c=2$).\n\nSawhney has provided the following proof that this series converges absolutely for $c>2$: note that, whenever $c>1$, the contribution to the sum from gaps $p_{n+1}-p_n\\geq \\log n$ is convergent, so it suffices to consider only small gaps. The number of $n\\leq X$ such that $p_{n+1}-p_n\\in [\\epsilon\\log n,2\\epsilon \\log n)$ is bounded above by $\\ll \\epsilon X$ (this can be proved via the Selberg sieve). In particular, applying this bound for $\\frac{1}{\\log n}\\leq \\epsilon \\leq 1$ of the shape $2^{-j}$ (of which there are at most $\\log\\log n$ possibilities) shows the desired convergence, since\\[\\sum \\frac{1}{n(\\log n)(\\log\\log n)^{c-1}}\\]converges. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#15: [Er97,p.158][Er97e,p.535][Er98]" }, { "number": 17, "url": "https://www.erdosproblems.com/17", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A038133" ], "formalized": "yes", "statement": "Are there infinitely many primes $p$ such that every even number $n\\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\\leq p$?", "commentary": "The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.\n\nBlecksmith, Erdős, and Selfridge [BES99] proved that the number of such primes is\\[\\ll_A \\frac{x}{(\\log x)^A}\\]for every $A>0$, and Elsholtz [El03] improved this to\\[\\ll x\\exp(-c(\\log\\log x)^2)\\]for every $c<1/8$.\n\nThis is discussed in problem C1 of Guy's collection [Gu04]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#17: [Er95,p.172]" }, { "number": 18, "url": "https://www.erdosproblems.com/18", "status": "open", "prize": "no", "tags": [ "number theory", "divisors", "factorials" ], "oeis": [ "A005153" ], "formalized": "yes", "statement": "We call $m$ practical if every integer $1\\leq nn$). He writes 'at the moment I do not even have a plausible conjecture, but perhaps this will not be hard to find'. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#19: [Er76b,p.171][Er76c,p.9][Er78,p.29][Er81][Er90][Er92b][Er93,p.341][Er95][Er97c][Er97d][Er97f][Va99,3.57]" }, { "number": 20, "url": "https://www.erdosproblems.com/20", "status": "open", "prize": "$1000", "tags": [ "combinatorics" ], "oeis": [ "A332077" ], "formalized": "yes", "statement": "Let $f(n,k)$ be minimal such that every family $\\mathcal{F}$ of $n$-uniform sets with $\\lvert \\mathcal{F}\\rvert \\geq f(n,k)$ contains a $k$-sunflower. Is it true that\\[f(n,k) < c_k^n\\]for some constant $c_k>0$?", "commentary": "Erdős and Rado [ErRa60] originally proved $f(n,k)\\leq (k-1)^nn!$. Kostochka [Ko97] improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erdős awarded him the consolation prize of \\$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang [ALWZ20] proved\\[f(n,k) < (Ck\\log n\\log\\log n)^n\\]for some constant $C>1$. This was refined slightly, independently by Rao [Ra20], Frankston, Kahn, Narayanan, and Park [FKNP19], and Bell, Chueluecha, and Warnke [BCW21], leading to the current record of\\[f(n,k) < (Ck\\log n)^n\\]for some constant $C>1$. This proof was streamlined by Hu. A constant of $C=64$ was achieved in the presentation by Stoeckl.\n\nIn [Er81] Erdős offered \\$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.\n\nThe usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, Rödl, and Talysheva [KRT99] have shown\\[f(n,k)=(1+O_n(k^{-1/2^n}))k^n.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#20: [Er65b][Er69][Er71,p.104][Er73][Er78,p.35][Er81][Er90][Er95][Er97c][Er97d][Va99,3.63]" }, { "number": 23, "url": "https://www.erdosproblems.com/23", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [ "A389646" ], "formalized": "yes", "statement": "Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?", "commentary": "The blow-up of $C_5$ shows that this would be the best possible. The best known bound is due to Balogh, Clemen, and Lidicky [BCL21], who proved that deleting at most $1.064n^2$ edges suffices.\n\nIn [Er92b] Erdős asks, more generally, if a graph on $(2k+1)n$ vertices in which every odd cycle has size $\\geq 2k+1$ can be made bipartite by deleting at most $n^2$ edges.\n\nThis problem is #58 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#23: [Er71][EFPS88][Er90][Er93,p.343][Er97b][Er97f]" }, { "number": 25, "url": "https://www.erdosproblems.com/25", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $1\\leq n_10$. \n\nAnother stronger conjecture would be that the hypothesis $\\lvert A\\cap [1,N]\\rvert \\gg N^{1/2}$ for all large $N$ suffices.\n\nSee also [40], and [1145] for a stronger generalisation.\n\nThis is discussed in problem C9 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#28: [ErTu41][Er56][Er57][Er59][Er61][Er65][Er65b][Er69][Er70c][Er73][Er77c][Er80,p.98][ErGr80][Er81][Er85c][Er89d][Er90][Er94b][Er95][Er97c][Er97f][Va99,1.16]" }, { "number": 30, "url": "https://www.erdosproblems.com/30", "status": "open", "prize": "$1000", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "A143824", "A227590", "A003022" ], "formalized": "yes", "statement": "Let $h(N)$ be the maximum size of a Sidon set in $\\{1,\\ldots,N\\}$. Is it true that, for every $\\epsilon>0$,\\[h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)?\\]", "commentary": "A problem of Erdős and Turán. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Both proofs in fact give\\[h(N) \\leq N^{1/2}+N^{1/4}+1.\\]Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to $0.998N^{1/4}$. This was further optimised by O'Bryant [OB22]. The current record is\\[h(N)\\leq N^{1/2}+0.98183N^{1/4}+O(1),\\]due to Carter, Hunter, and O'Bryant [CHO25].\n\nSinger [Si38] was the first to show that $h(N)\\geq (1-o(1))N^{1/2}$ for all $N$. For a detailed survey of the literature we refer to [OB04].\n\nSee also [241] and [840].\n\nThis problem is Problem 31 on Green's open problems list.\n\nThis is discussed in problem C9 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#30: [Er61][Er69][Er70b][Er70c][Er72][Er73][Er77c][Er80,p.99][Er80e][Er81][Er81h,p.174][Er91][Er92c][Er94b][Er95][Er97c][Va99,1.18]" }, { "number": 32, "url": "https://www.erdosproblems.com/32", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "yes", "statement": "Is there a set $A\\subset\\mathbb{N}$ such that\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert = o((\\log N)^2)\\]and such that every large integer can be written as $p+a$ for some prime $p$ and $a\\in A$? Can the bound $O(\\log N)$ be achieved? Must such an $A$ satisfy\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}> 1?\\]", "commentary": "Such a set is called an additive complement to the primes.\n\nErdős [Er54] proved that such a set $A$ exists with $\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\ll (\\log N)^2$ (improving a previous result of Lorentz [Lo54] who achieved $\\ll (\\log N)^3$). \n\nWolke [Wo96] has shown that such a bound is almost true, in that we can achieve $\\ll (\\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable. Kolountzakis [Ko96] improved this to $\\ll (\\log N)(\\log\\log N)$, and Ruzsa [Ru98c] further improved this to $\\ll \\omega(N)\\log N$ for any $\\omega\\to \\infty$.\n\nThe answer to the third question is yes: Ruzsa [Ru98c] has shown that we must have\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}\\geq e^\\gamma\\approx 1.781.\\]This is discussed in problem E1 of Guy's collection [Gu04], where it is stated that Erdős offered \\$50 for determining whether $O(\\log N)$ can be achieved.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#32: [Er56,p.132][Er57,p.295][Er59,p.117][Er61,p.229][Er65b,p.227][Er73,p.133][Er77c,p.62][Va99,1.9]" }, { "number": 33, "url": "https://www.erdosproblems.com/33", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of\\[\\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}?\\]Is\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1?\\]", "commentary": "Such a set $A$ is called an additive complement of the set of squares. Erdős observed that there exist $A$ for which the $\\limsup$ is finite and $>1$. Moser [Mo65] proved that, for any such $A$,\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1.06.\\]The best-known lower bound is\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\geq\\frac{4}{\\pi}\\approx 1.273\\]proved by Cilleruelo [Ci93], Habsieger [Ha95], and Balasubramanian and Ramana [BaRa01].\n\nThe problem of minimising the $\\limsup$ appears to have been much less studied. van Doorn has a construction of such an $A$ in which, for all $N$,\\[\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}< 2\\phi^{5/2}\\approx 6.66,\\]where $\\phi=\\frac{1+\\sqrt{5}}{2}$ is the golden ratio.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#33: [Er56,p.134]" }, { "number": 36, "url": "https://www.erdosproblems.com/36", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "A393584" ], "formalized": "yes", "statement": "Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\\sqcup B=\\{1,\\ldots,2N\\}$ is a partition into two equal parts, so that $\\lvert A\\rvert=\\lvert B\\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\\in A$ and $b\\in B$ is at least $cN$.", "commentary": "The minimum overlap problem. The example (with $N$ even) $A=\\{N/2+1,\\ldots,3N/2\\}$ shows that $c\\leq 1/2$ (indeed, Erdős initially conjectured that $c=1/2$). The lower bound of $c\\geq 1/4$ is trivial, and Scherk improved this to $1-1/\\sqrt{2}=0.29\\cdots$. The current records are\\[0.379005 < c < 0.380876,\\]the lower bound due to White [Wh22] and the upper bound due to the TTT-Discover LLM [YKLBMWKCZGS26], improving slightly on earlier bounds due to AlphaEvolve [GGTW25] and Haugland [Ha16].\n\nThis is discussed in problem C17 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#36: [Er55][Er56][Er61][Er92c]" }, { "number": 38, "url": "https://www.erdosproblems.com/38", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that\\[\\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N\\]where $f(\\alpha)>0$ for $0<\\alpha <1 $?The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]", "commentary": "Erdős [Er36c] proved that if $B$ is an additive basis of order $k$ then, for any set $A$ of Schnirelmann density $\\alpha$, for every $N$ there exists some integer $b\\in B$ such that\\[\\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq \\left(\\alpha+\\frac{\\alpha(1-\\alpha)}{2k}\\right)N.\\]It seems an interesting question (not one that Erdős appears to have asked directly, although see [35]) to improve the lower bound here, even in the case $B=\\mathbb{N}$. Erdős observed that a random set of density $\\alpha$ shows that the factor of $\\frac{\\alpha(1-\\alpha)}{2}$ in this case cannot be improved past $\\alpha(1-\\alpha)$.\n\nThis is a stronger property than $B$ being an essential component (see [37]). Linnik [Li42] gave the first construction of an essential component which is not an additive basis.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 September 2025. View history", "references": "#38: [Er56,p.136]" }, { "number": 39, "url": "https://www.erdosproblems.com/39", "status": "open", "prize": "$500", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that\\[\\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}\\]for all $\\epsilon>0$?", "commentary": "The trivial greedy construction achieves $\\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Komlós, and Szemerédi [AKS81b], who found an infinite Sidon set with growth rate $\\gg (N\\log N)^{1/3}$. The current best bound of $\\gg N^{\\sqrt{2}-1+o(1)}$ is due to Ruzsa [Ru98]. \n\nErdős [Er73] had offered \\$25 for any construction which achieves $N^{c}$ for some $c>1/3$. Later he ([Er77c] and [Er80]) offered \\$100 for a construction which achieves $\\omega(N)N^{1/3}$ for some $\\omega(N)\\to \\infty$.\n\nErdős proved that for every infinite Sidon set $A$ we have\\[\\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0.\\]Erdős and Rényi have constructed, for any $\\epsilon>0$, a set $A$ such that\\[\\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}\\]for all large $N$ and $1_A\\ast 1_A(n)\\ll_\\epsilon 1$ for all $n$.\n\nThis is discussed in problem C9 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#39: [Er56][Er61][Er73][Er77c][Er80,p.98][ErGr80,p.48][Er81][Er82e][Er85c][Er91][Er95][Er97c][Va99,1.18]" }, { "number": 40, "url": "https://www.erdosproblems.com/40", "status": "open", "prize": "$500", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "yes", "statement": "For what functions $g(N)\\to \\infty$ is it true that\\[\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg \\frac{N^{1/2}}{g(N)}\\]implies $\\limsup 1_A\\ast 1_A(n)=\\infty$?", "commentary": "This is a stronger form of the Erdős-Turán conjecture [28] (since establishing this for any function $g(N)\\to \\infty$ would imply a positive solution to [28]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#40: [Er95][Er97c]" }, { "number": 41, "url": "https://www.erdosproblems.com/41", "status": "open", "prize": "$500", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset\\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\\in A$ (aside from the trivial coincidences). Is it true that\\[\\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/3}}=0?\\]", "commentary": "Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then\\[\\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0.\\]This is discussed in problem C11 of Guy's collection [Gu04], in which Guy says Erdős offered \\$500 for the general problem of whether, for all $h\\geq 2$,\\[\\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/h}}=0\\]whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash [Na89] and for all even $h$ by Chen [Ch96b]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#41: [Er77c][Er80,p.99][ErGr80][Er81][Er85c][Er91][Er95][Er97c][Va99,1.23]" }, { "number": 42, "url": "https://www.erdosproblems.com/42", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", "commentary": "Sedov in the comments (using ChatGPT and Codex) has proved this is true for $M=3$. The case $M=1$ is trivial; the case $M=2$ is a little less trivial, but is also proved by Sedov in the comments.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#42: [Er95]" }, { "number": 43, "url": "https://www.erdosproblems.com/43", "status": "open", "prize": "$100", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "A143824", "A227590", "A003022" ], "formalized": "no", "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?", "commentary": "Since it is known that $f(N)\\sim \\sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\\lvert A\\rvert=\\lvert B\\rvert$,\\[\\lvert A\\rvert \\leq \\left(\\frac{1}{\\sqrt{2}}-c+o(1)\\right)\\sqrt{N}\\]for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.\n\nIn the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\\subset \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lvert B\\rvert$ and $(A-A)\\cap (B-B)=\\{0\\}$ and\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\geq (1-o(1))\\binom{f(N)}{2}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 December 2025. View history", "references": "#43: [Er95]" }, { "number": 44, "url": "https://www.erdosproblems.com/44", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may depend on $N,A,\\epsilon$) such that $A\\cup B\\subset \\{1,\\ldots,M\\}$ is a Sidon set of size at least $(1-\\epsilon)M^{1/2}$?", "commentary": "See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).\n\nThis is discussed in problem C9 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 09 January 2026. View history", "references": "#44: [Er84b,p.16][Er91][Er95][Er97c]" }, { "number": 50, "url": "https://www.erdosproblems.com/50", "status": "open", "prize": "$250", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Schoenberg proved that for every $c\\in [0,1]$ the density of\\[\\{ n\\in \\mathbb{N} : \\phi(n)0$\\[\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}?\\]", "commentary": "The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\\lvert A\\rvert^{1+c}$ for some constant $c>0$, and an upper bound of\\[\\lvert A\\rvert^2 \\exp\\left(-c\\frac{\\log\\lvert A\\rvert}{\\log\\log \\lvert A\\rvert}\\right)\\]for some constant $c>0$. The lower bound has been improved a number of times. The current record is\\[\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{1962}{1469}-o(1)}\\]due to Cushman [Cu25] (note $1962/1469=1.3356\\cdots$). A complete history of sum-product bounds can be found at this webpage.\n\nThere is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Bloom above. The best bound for complex numbers is\\[\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{4}{3}+c}\\]for some absolute constant $c>0$, due to Basit and Lund [BaLu19]. \n\n\nOne can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that there exists $c>0$ such that if $A\\subseteq \\mathbb{F}_p$ with $\\lvert A\\rvert \\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many copies of $C_4$?", "commentary": "Conjectured by Erdős and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.\n\nThe behaviour of $\\mathrm{ex}(n;C_4)$ is the subject of [765].\n\nHe, Ma, and Yang [HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 November 2025. View history", "references": "#60: [Er90][Er93,p.335]" }, { "number": 61, "url": "https://www.erdosproblems.com/61", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "yes", "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\\geq n^c$ vertices?", "commentary": "Conjectured by Erdős and Hajnal [ErHa89], who proved that a complete graph or independent set must exist on\\[\\geq \\exp(c_H\\sqrt{\\log n})\\]many vertices, where $c_H>0$ is some constant. This was improved by Bucić, Nguyen, Scott, and Seymour [BNSS23] to\\[\\geq \\exp(c_H\\sqrt{\\log n\\log\\log n}).\\]The conjecture has been proved:\n for $H$ with $\\leq 4$ vertices by Erdős and Hajnal [ErHa89].\n for the 'bull' graph on $5$ vertices (a $P_4$ with a new vertex adjacent to the two middle vertices) by Chudnovsky and Safra [ChSa08].\n for $C_5$ by Chudnovsky, Scott, Seymour, and Spirkl [CSSS23].\n for $P_5$ by Nguyen, Scott, and Seymour [NSS26].\nAlon, Pach, and Solymosi [APS01] have proved that the family of $H$ for which the conjecture holds is closed under vertex substitution; together with the three final results above this means the conjecture is now known for all $H$ on $5$ vertices.\n\nNguyen, Scott, and Seymour [NSS24] have proved that if $H$ is a path then all $H$-free graphs on $n$ vertices contain either a complete graph or independent set on\\[\\geq 2^{(\\log n)^{1-o(1)}}\\]many vertices.\n\nA detailed account of this problem, including the proof of some special cases, is given in the PhD thesis of Nguyen. \n\nThis problem is #80 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#61: [ErHa89][Er90][Er93,p.346][Er97f][Va99,3.52]" }, { "number": 62, "url": "https://www.erdosproblems.com/62", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "If $G_1,G_2$ are two graphs with chromatic number $\\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?", "commentary": "Erdős also asked [Er87] about finding a common subgraph $H$ (with chromatic number either $4$ or $\\aleph_0$) in any finite collection of graphs with chromatic number $\\aleph_1$. \n\nEvery graph with chromatic number $\\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594]. This was proved by Erdős, Hajnal, and Shelah [EHS74]. Erdős wrote [Er87] that 'probably' every graph with chromatic number $\\aleph_1$ contains as subgraphs all graphs with chromatic number $4$ with sufficiently large girth.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#62: [Er87][Er90][Er95d][Va99,7.89]" }, { "number": 64, "url": "https://www.erdosproblems.com/64", "status": "falsifiable", "prize": "$1000", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "yes", "statement": "Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\\geq 2$?", "commentary": "Conjectured by Erdős and Gyárfás, who believed the answer must be negative, and in fact for every $r$ there must be a graph of minimum degree at least $r$ without a cycle of length $2^k$ for any $k\\geq 2$.\n\nThis was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove a much stronger result: if the average degree of $G$ is sufficiently large then there is some large integer $\\ell$ such that for every even integer $m\\in [(\\log \\ell)^8,\\ell]$, $G$ contains a cycle of length $m$.\n\nAn infinite tree with minimum degree $3$ shows that the answer is trivially false for infinite graphs.\n\nThe conjecture has been confirmed for various families of graphs; see the comment by Alfaiz for a list.\n\nThis problem is #69 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#64: [Er93,p.343][Er94b][Er95,p.174][Er96][Er97b][Er97c]" }, { "number": 65, "url": "https://www.erdosproblems.com/65", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_10$.\n\nIt is open even for $f(n)=\\sqrt{n}$. Erdős offered \\$500 for a proof but only \\$250 for a counterexample. This fails (even with $f(n)\\gg n$) if the graph has chromatic number $\\aleph_1$ (see [111]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 25 January 2026. View history", "references": "#74: [EHS82][Er87][Er90][Er93,p.342][Er94b][Er95][Er95d,p.62][Er96][Er97b][Er97c][Er97d][Er97f]" }, { "number": 75, "url": "https://www.erdosproblems.com/75", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Is there a graph of chromatic number $\\aleph_1$ with $\\aleph_1$ vertices such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\\epsilon}$? What about an independent set of size $\\gg n$?", "commentary": "Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].\n\nIn [Er95] Erdős asks this without the condition that the graph also have $\\aleph_1$ vertices, but this is an oversight, since already in [EHS82] they provide such a construction.\n\nIn [Er95d] Erdős offers \\$1000 for a complete solution to all problems of this type (for example including also [74]), and a 'generous reward for any significant partial results'.\n\nSee also [750].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 January 2026. View history", "references": "#75: [EHS82,p.120][Er95,p.11][Er95d,p.63]" }, { "number": 77, "url": "https://www.erdosproblems.com/77", "status": "open", "prize": "$250", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442" ], "formalized": "no", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of\\[\\lim_{k\\to \\infty}R(k)^{1/k}.\\]", "commentary": "Erdős offered \\$100 for just a proof of the existence of this constant, without determining its value. He also offered \\$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In [Er88] he raises this prize to \\$10000). Erdős proved\\[\\sqrt{2}\\leq \\liminf_{k\\to \\infty}R(k)^{1/k}\\leq \\limsup_{k\\to \\infty}R(k)^{1/k}\\leq 4.\\]The upper bound has been improved to $4-\\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CGMS23]. This was improved to $3.7992\\cdots$ by Gupta, Ndiaye, Norin, and Wei [GNNW24].\n\nA shorter and simpler proof of an upper bound of the strength $4-c$ for some constant $c>0$ (and a generalisation to the case of more than two colours) was given by Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba [BBCGHMST24].\n\nIn [Er93] Erdős writes 'I have no idea what the value of $\\lim R(k)^{1/k}$ should be, perhaps it is $2$ but we have no real evidence for this.'\n\nThis problem is #3 in Ramsey Theory in the graphs problem collection.\n\nSee also [1029] for a problem concerning a lower bound for $R(k)$ and discussion of lower bounds in general. The limit in this question is closely related to the limit in [627].\n\nA famous quote of Erdős concerns the difficulty of finding exact values for $R(k)$. This is often repeated in the words of Spencer, who phrased it as an alien attacking race. The earliest such quote in a paper of Erdős I have found is in [Er93], where he writes:\n\n'Sometime ago, I made the following joke. If an evil spirit would appear and say \"unless you give me the value of $R(5)$ within a year, I will exterminate humanity\", then our best bet would be perhaps to get all our computers working on $R(5)$ and we probably would get its value in a year. \n\nIf he would ask for $R(6)$, the best strategy probably would be to destroy it before it can destroy us. If we would be so clever that we could give the answer by mathematics, we would just tell him: \"if you try to do something you will see what will happent to you...\". I think we are strong enugh now and the only evil spirit we have to feel is the one which is in ourselves (quoting somebody: I have seen the enemy and them are us). Now enough of the idle talk and back to Mathematics.'\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#77: [Er61][Er69b][Er71,p.99][Er81][Er88,p.83][Er90b,p.17][Er93,p.338][Er95][Er97c][Er97d][Va99,3.50]" }, { "number": 78, "url": "https://www.erdosproblems.com/78", "status": "open", "prize": "$100", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442" ], "formalized": "no", "statement": "Let $R(k)$ be the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$.Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.", "commentary": "Erdős gave a simple probabilistic proof that $R(k) \\gg k2^{k/2}$. \n\nEquivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\\geq c\\log n$ for some constant $c>0$. \n\nIn [Er69b] Erdős asks for even a construction whose largest clique or independent set has size $o(n^{1/2})$, which is now known.\n\n Cohen [Co15] (see the introduction for further history) constructed a graph on $n$ vertices which does not contain any clique or independent set of size\\[\\geq 2^{(\\log\\log n)^{C}}\\]for some constant $C>0$. Li [Li23b] has recently improved this to\\[\\geq (\\log n)^{C}\\]for some constant $C>0$.\n\nThis problem is #4 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#78: [Er69b][Er71][Er88][Er93,p.337][Er95][Er97c][Va99,3.49]" }, { "number": 80, "url": "https://www.erdosproblems.com/80", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles. Estimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\\epsilon}$ for some $\\epsilon>0$? Or $f_c(n)\\gg \\log n$?", "commentary": "A problem of Erdős and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\\ll_c n^{1/2}$. Szemerédi observed that his regularity lemma implies that $f_c(n)\\to \\infty$.\n\nEdwards (unpublished) and Khadzhiivanov and Nikiforov [KhNi79] proved independently that $f_c(n) \\geq n/6$ when $c>1/4$ (see [905]).\n\nFox and Loh [FoLo12] proved that\\[f_c(n) \\leq n^{O(1/\\log\\log n)}\\]for all $c<1/4$, disproving the first conjecture of Erdős.\n\nThe best known lower bounds for $f_c(n)$ are those from Szemerédi's regularity lemma, and as such remain very poor.\n\nSee also [600] and the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#80: [Er87]" }, { "number": 81, "url": "https://www.erdosproblems.com/81", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?", "commentary": "Asked by Erdős, Ordman, and Zalcstein [EOZ93], who proved an upper bound of $(1/4-\\epsilon)n^2$ many cliques (for some very small $\\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.\n\nA split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\\frac{3}{16}n^2+O(n)$ many cliques.\n\nSee also [1017].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#81: [Er95]" }, { "number": 82, "url": "https://www.erdosproblems.com/82", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [ "A120414", "A390256", "A390257", "A390919", "A392636", "A394400", "A394462", "A394539", "A394563", "A394564", "A394573", "A394574", "A394930", "A394933" ], "formalized": "yes", "statement": "Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\\log n\\to \\infty$.", "commentary": "Conjectured by Erdős, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$. \n\nRamsey's theorem implies that $F(n)\\gg \\log n$. Bollobás observed that $F(n)\\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov [AKS07] have improved this to $n^{1/2}(\\log n)^{O(1)}$. Dyson and McKay [DyMc26] have improved this further to $F(n) \\ll n^{1/2}$. \n\nIn [Er93] Erdős asks whether, if $t(n)$ is the largest trivial (either empty or complete) subgraph which a graph on $n$ vertices must contain (so that $t(n) \\gg \\log n$ by Ramsey's theorem), then is it true that\\[F(n)-t(n)\\to \\infty?\\]Equivalently, and in analogue with the definition of Ramsey numbers, one can define $G(n)$ to be the minimal $m$ such that every graph on $m$ vertices contains a regular induced subgraph on at least $n$ vertices. This problem can be rephrased as asking whether $G(n) \\leq 2^{o(n)}$. \n\nFajtlowicz, McColgan, Reid, and Staton [FMRS95] showed that $G(1)=1$, $G(2)=2$, $G(3)=5$, $G(4)=7$, and $G(5)\\geq 12$. Boris Alexeev and Brendan McKay (see the comments and this site) have computed $G(5)=17$, $G(6)\\geq 21$, and $G(7)\\geq 29$. Dyson and McKay [DyMc26] have proved $G(7)\\geq 30$, and $G(k)\\geq \\frac{9}{163}k^2$ for all large $k$.\n\nSee also [1031] for another question regarding induced regular subgraphs.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#82: [Er93,p.340][Er95][Er97d]" }, { "number": 84, "url": "https://www.erdosproblems.com/84", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "no", "statement": "The cycle set of a graph $G$ on $n$ vertices is a set $A\\subseteq \\{3,\\ldots,n\\}$ such that there is a cycle in $G$ of length $\\ell$ if and only if $\\ell \\in A$. Let $f(n)$ count the number of possible such $A$. Prove that $f(n)=o(2^n)$.Prove that $f(n)/2^{n/2}\\to \\infty$.", "commentary": "Conjectured by Erdős and Faudree, who showed that $2^{n/2}f(n)-c$ for all $m>n$. This question can be asked for other graphs than $C_4$.\n\nThe bounds in [552] imply in particular that $f(n)<\\sqrt{n}+1$ and\\[f(n)=(1+o(1))\\sqrt{n}.\\]It is easy to check that $f(4)=2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#85: [Er93,p.345][Er94b][Er95][Er96]" }, { "number": 86, "url": "https://www.erdosproblems.com/86", "status": "open", "prize": "$100", "tags": [ "graph theory" ], "oeis": [ "A245762" ], "formalized": "no", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with\\[\\geq \\left(\\frac{1}{2}+o(1)\\right)n2^{n-1}\\]many edges contains a $C_4$?", "commentary": "Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\\leq (\\frac{1}{2}+o(1))n2^{n-1}$.\n\nErdős [Er91] showed that\\[f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{n}\\right)n2^{n-1}\\]for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg [BHN95] improved this to\\[f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{\\sqrt{n}}\\right)n2^{n-1}\\]for some constant $c>0$.\n\nBalogh, Hu, Lidicky, and Liu [BHLL14] proved that $f(n)\\leq 0.6068 n2^{n-1}$. This was improved to $\\leq 0.60318 n2^{n-1}$ by Baber [Ba12b].\n\nA similar question can be asked for other even cycles.\n\nSee also [666] and the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#86: [Er90][Er91][Er92b][Er93,p.343][Er94b][Er95][Er97f]" }, { "number": 87, "url": "https://www.erdosproblems.com/87", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442" ], "formalized": "no", "statement": "Let $\\epsilon >0$. Is it true that, if $k$ is sufficiently large, then\\[R(G)>(1-\\epsilon)^kR(k)\\]for every graph $G$ with chromatic number $\\chi(G)=k$? Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\\chi(G)=k$?", "commentary": "Erdős originally conjectured that $R(G)\\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay [FaMc93] showed that $R(W)=17$ for the pentagonal wheel $W$.\n\nSince $R(k)\\leq 4^k$ this is trivial for $\\epsilon\\geq 3/4$. Yuval Wigderson points out that $R(G)\\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.\n\nThis problem is #12 and #13 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 17 January 2026. View history", "references": "#87: [Er95,p.14]" }, { "number": 89, "url": "https://www.erdosproblems.com/89", "status": "open", "prize": "$500", "tags": [ "geometry", "distances" ], "oeis": [ "A186704", "A131628" ], "formalized": "yes", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ determine $\\gg n/\\sqrt{\\log n}$ many distinct distances?", "commentary": "A $\\sqrt{n}\\times\\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\\gg n/\\log n$ many distinct distances. \n\nA stronger form (see [604]) may be true: is there a single point which determines $\\gg n/\\sqrt{\\log n}$ distinct distances, or even $\\gg n$ many such points, or even that this is true averaged over all points - for example, if $d(x)$ counts the number of distinct distances from $x$ then in [Er75f] Erdős conjectured\\[\\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}},\\]where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\n\nSee also [661], and [1083] for the generalisation to higher dimensions.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#89: [Er46b][Er57][Er61][Er75f,p.99][Er81][Er82e][Er83c][Er85][Er87b,p.170][Er90][Er92e][Er95][Er97b][Er97c][Er97e][Er97f][Va99,4.69]" }, { "number": 90, "url": "https://www.erdosproblems.com/90", "status": "open", "prize": "$500", "tags": [ "geometry", "distances" ], "oeis": [ "A186705" ], "formalized": "yes", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", "commentary": "The unit distance problem. In [Er94b] Erdős dates this conjecture to 1946. In [Er82e] he offers \\$300 for the upper bound $n^{1+o(1)}$.\n\nThis would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemerédi, and Trotter [SST84]. In [Er83c] and [Er85] Erdős offers \\$250 for an upper bound of the form $n^{1+o(1)}$. \n\nPart of the difficulty of this problem is explained by a result of Valtr (see [Sz16]), who constructed a metric on $\\mathbb{R}^2$ and a set of $n$ points with $\\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemerédi, and Trotter [SST84] generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.\n\nSee a survey by Szemerédi [Sz16] for further background and related results.\n\nSee also [92], [96], [605], and [956]. The higher dimensional generalisation is [1085].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#90: [Er46b][Er61][Er75f,p.100][Er81][Er82e][Er83c][Er85][Er90][Er94b][Er95][Er97c][Er97e][Er97f][Va99,4.67]" }, { "number": 91, "url": "https://www.erdosproblems.com/91", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [ "A186704" ], "formalized": "no", "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "commentary": "For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.\n\nFor $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erdős mysteriously remarks in [Er90] this was proved by 'a colleague'. (In [Er87b] this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kovács [Ko24c].\n\nIn [Er87b] Erdős says that there are at least two non-similar examples for $6\\leq n\\leq 9$.\n\nThe minimal possible number of distinct distances is the subject of [89].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 January 2026. View history", "references": "#91: [Er87b,p.171][Er90][Er97e]" }, { "number": 92, "url": "https://www.erdosproblems.com/92", "status": "open", "prize": "$500", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.Is it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", "commentary": "This is a stronger form of the unit distance conjecture (see [90]).\n\nThe set of lattice points imply $f(n) > n^{c/\\log\\log n}$ for some constant $c>0$. Erdős offered \\$500 for a proof that $f(n) \\leq n^{o(1)}$ but only \\$100 for a counterexample. This latter prize is downgraded to \\$50 in [ErFi97].\n\nIt is trivial that $f(n) \\ll n^{1/2}$. A result of Pach and Sharir (Theorem 4 of [PaSh92]) implies $f(n) \\ll n^{2/5}$. Hunter has observed that the circle-point incidence bound of Janzer, Janzer, Methuku, and Tardos [JJMT24] implies\\[f(n) \\ll n^{4/11}.\\]Fishburn (personal communication to Erdős, later published in [ErFi97]) proved that $6$ is the smallest $n$ such that $f(n)=3$ and $8$ is the smallest $n$ such that $f(n)=4$, and suggested that the lattice points may not be best example.\n\nSee also [754].\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#92: [Er75f,p.100][Er94b][Er95,p.180][Er97c,p.65]" }, { "number": 96, "url": "https://www.erdosproblems.com/96", "status": "open", "prize": "no", "tags": [ "geometry", "distances", "convex" ], "oeis": [], "formalized": "no", "statement": "If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.", "commentary": "Conjectured by Erdős and Moser. In [Er92e] Erdős credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. Füredi [Fu90] proved an upper bound of $O(n\\log n)$. A short proof of this bound was given by Brass and Pach [BrPa01]. The best known upper bound is\\[\\leq n\\log_2n+4n,\\]due to Aggarwal [Ag15].\n\nEdelsbrunner and Hajnal [EdHa91] have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erdős and Moser, that the true answer was $\\frac{5}{3}n+O(1)$.)\n\nA positive answer would follow from [97]. See also [90].\n\nIn [Er92e] Erdős makes the stronger conjecture that, if $g(x)$ counts the largest number of points equidistant from $x$ in $A$, then\\[\\sum_{x\\in A}g(x)< 4n.\\]He notes that the example of Edelsbrunner and Hajnal shows that $\\sum_{x\\in A}g(x)>4n-O(1)$ is possible.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#96: [Er90][Er92e][Er97e][Er97f][Va99,4.68]" }, { "number": 97, "url": "https://www.erdosproblems.com/97", "status": "falsifiable", "prize": "$100", "tags": [ "geometry", "distances", "convex" ], "oeis": [], "formalized": "yes", "statement": "Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?", "commentary": "Erdős originally conjectured this (in [Er46b]) with no $3$ vertices equidistant, but Danzer found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this distance depends on the vertex). Danzer's construction is explained in [Er87b]. Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices).\n\nIf this fails for $4$, perhaps there is some constant for which it holds? In [Er75f] Erdős claimed that Danzer proved that this false for every constant - in fact, for any $k$ there is a convex polygon such that every vertex has $k$ vertices equidistant from it. Since this claim was not repeated in later papers, presumably Erdős was mistaken here.\n\nErdős suggested this as an approach to solve [96]. Indeed, if this problem holds for $k+1$ vertices then, by induction, this implies an upper bound of $kn$ for [96].\n\nThe answer is no if we omit the requirement that the polygon is convex (I thank Boris Alexeev and Dustin Mixon for pointing this out), since for any $d$ there are graphs with minimum degree $d$ which can be embedded in the plane such that each edge has length one (for example one can take the $d$-dimensional hypercube graph on $2^d$ vertices). One can then connect the vertices in a cyclic order so that there are no self-intersections and no three consecutive vertices on a line, thus forming a (non-convex) polygon.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 October 2025. View history", "references": "#97: [Er46b][Er61][Er75f,p.100][Er87b,p.175][Er90][Er92e][Er95][Er97e]" }, { "number": 98, "url": "https://www.erdosproblems.com/98", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\\to \\infty$?", "commentary": "Erdős could not even prove $h(n)\\geq n$. Pach has shown $h(n)0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 15 October 2025. View history", "references": "#98: [Er75f,p.101][Er83c][Er87b,p.167][Er90][Er92b][EFPR93][Er94b][Er97e]" }, { "number": 99, "url": "https://www.erdosproblems.com/99", "status": "open", "prize": "$100", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq\\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?", "commentary": "Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erdős believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).\n\nErdős [Er94b] wrote 'I could not prove it but felt that it should not be hard. To my great surprise both B. H. Sendov and M. Simonovits doubted the truth of this conjecture.' In [Er94b] he offers \\$100 for a counterexample but only \\$50 for a proof.\n\nThe stated problem is false for $n=4$, for example taking the points to be vertices of a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor [BeFo99].\n\nSee also [103].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#99: [Er94b][Er95][Er97e]" }, { "number": 100, "url": "https://www.erdosproblems.com/100", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $A$ be a set of $n$ points in $\\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\\gg n$?", "commentary": "Perhaps the diameter is even $\\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz [GuKa15] imply a lower bound of $\\gg n/\\log n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#100: [Er90][Er92e][Er95][Er97f]" }, { "number": 101, "url": "https://www.erdosproblems.com/101", "status": "open", "prize": "$100", "tags": [ "geometry" ], "oeis": [ "A006065" ], "formalized": "no", "statement": "Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.", "commentary": "There are examples of sets of $n$ points with $\\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84]. \n\nGrünbaum [Gr76] constructed an example with $\\gg n^{3/2}$ such lines. Erdős speculated this may be the correct order of magnitude. This is false: Solymosi and Stojaković [SoSt13] have constructed a set with no five on a line and at least\\[n^{2-O(1/\\sqrt{\\log n})}\\]many lines containing exactly four points.\n\nSee also [102] and [669]. A generalisation of this problem is asked in [588].\n\nThis problem is Problem 71 on Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#101: [Er84][Er87b,p.170][Er90][Er92e][Er95,p.181][Er97c,p.66]" }, { "number": 102, "url": "https://www.erdosproblems.com/102", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\\to \\infty$?", "commentary": "A problem of Erdős and Purdy. It is not even known if $h_c(n)\\geq 5$ (see [101]). \n\nIt is easy to see that $h_c(n) \\ll_c n^{1/2}$, and Erdős at one point [Er95] suggested that perhaps a similar lower bound $h_c(n)\\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\\{1,\\ldots,m\\}^d$ where $n\\approx m^d$. These intersect any line in $\\ll_d n^{1/d}$ points, and have $\\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\\mathbb{R}^d$, but a random projection into $\\mathbb{R}^2$ preserves the relevant properties.\n\nThis construction shows that $h_c(n) \\ll n^{1/\\log(1/c)}$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#102: [Er92e][Er95][Er97c]" }, { "number": 103, "url": "https://www.erdosproblems.com/103", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ count the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\\geq 1$ for all points $x\\neq y$. Is it true that $h(n)\\to \\infty$?", "commentary": "It is not even known whether $h(n)\\geq 2$ for all large $n$.\n\nSee also [99].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#103: [Er94b]" }, { "number": 104, "url": "https://www.erdosproblems.com/104", "status": "open", "prize": "$100", "tags": [ "geometry" ], "oeis": [ "A003829" ], "formalized": "no", "statement": "Given $n$ points in $\\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.", "commentary": "In [Er81d] Erdős proved that $\\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erdős claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson [HaMe86] note that in fact this delivers an upper bound of $\\frac{n(n-1)}{3}$. \n\nElekes [El84] has a simple construction of a set with $\\gg n^{3/2}$ such circles. This may be the correct order of magnitude.\n\nIn [Er75h] and [Er92e] Erdős also asks how many such unit circles there must be if the points are in general position.\n\nIn [Er92e] Erdős offered £100 for a proof or disproof that the answer is $O(n^{3/2})$.\n\nThe maximal number of unit circles achieved by $n$ points is A003829 in the OEIS.\n\nSee also [506] and [831].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#104: [Er75h,p.2][Er81d,p.144][Er83b][Er92e,p.46][Er95,p.182]" }, { "number": 106, "url": "https://www.erdosproblems.com/106", "status": "falsifiable", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the squares. Is $f(k^2+1)=k$?", "commentary": "In [Er94b] Erdős dates this conjecture to 'more than 60 years ago'. Erdős proved that $f(2)=1$ in an early mathematical paper for high school students in Hungary. Newman proved (in personal communication to Erdős) that $f(5)=2$.\n\nIt is trivial from the Cauchy-Schwarz inequality that $f(k^2)=k$. Erdős also asks for which $n$ is it true that $f(n+1)=f(n)$.\n\nIt is easy to see that $f(k^2+1)\\geq k$, by first dividing the unit square into $k^2$ smaller squares of side-length $1/k$, and then replacing one square by two smaller squares of side-length $1/2k$. Halász [Ha84] gives a construction that shows $f(k^2+2)\\geq k+\\frac{1}{k+1}$, and in general, for any $c\\geq 1$,\\[f(k^2+2c+1)\\geq k+\\frac{c}{k}\\]and\\[f(k^2+2c)\\geq k+\\frac{c}{k+1}.\\]Halász also considers the variants where we replace a square by a parallelogram or triangle.\n\nErdős and Soifer [ErSo95] and Campbell and Staton [CaSt05] have conjectured that, in general, for any integer $-kk$) is also open.\n\nIn [Er79b] Erdős also asks whether\\[\\lim_{k\\to \\infty}\\frac{f(k,r+1)}{f(k,r)}=\\infty.\\]See also the entry in the graphs problem collection and [740] for the infinitary version.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#108: [Er71][Er79b][Er81][Er90][Er95d][Va99,3.59]" }, { "number": 111, "url": "https://www.erdosproblems.com/111", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number", "set theory" ], "oeis": [], "formalized": "no", "statement": "If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges. What is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\\to \\infty$ for every graph $G$ with chromatic number $\\aleph_1$?", "commentary": "A problem of Erdős, Hajnal, and Szemerédi [EHS82]. Every $G$ with chromatic number $\\aleph_1$ must have $h_G(n)\\gg n$ since $G$ must contain, for some $r$, $\\aleph_1$ many vertex disjoint odd cycles of length $2r+1$. \n\nOn the other hand, Erdős, Hajnal, and Szemerédi proved that there is a $G$ with chromatic number $\\aleph_1$ such that $h_G(n)\\ll n^{3/2}$. In [Er81] Erdős conjectured that this can be improved to $\\ll n^{1+\\epsilon}$ for every $\\epsilon>0$. \n\nSee also [74].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#111: [Er81][EHS82][Er87][Er90][Er97d][Er97f]" }, { "number": 112, "url": "https://www.erdosproblems.com/112", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.", "commentary": "A problem of Erdős and Rado [ErRa67], who showed $k(n,m) \\ll_m n^{m-1}$, or more precisely,\\[k(n,m) \\leq \\frac{2^{m-1}(n-1)^m+n-2}{2n-3}.\\]Larson and Mitchell [LaMi97] improved the dependence on $m$, establishing in particular that $k(n,3)\\leq n^{2}$. Zach Hunter has observed that\\[R(n,m) \\leq k(n,m)\\leq R(n,m,m),\\]which in particular proves the upper bound $k(n,m)\\leq 3^{n+2m}$.\n\n\nSee also the entry in the graphs problem collection - on this site the problem replaces transitive tournament with directed path, but Zach Hunter and Raphael Steiner have a simple argument that proves, for this alternative definition, that $k(n,m)=(n-1)(m-1)$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#112: [ErRa67]" }, { "number": 114, "url": "https://www.erdosproblems.com/114", "status": "falsifiable", "prize": "$250", "tags": [ "polynomials", "analysis" ], "oeis": [], "formalized": "no", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "commentary": "A problem of Erdős, Herzog, and Piranian [EHP58]. It is also listed as Problem 4.10 in [Ha74], where it is attributed to Erdős.\n\nIn [Va99] it is just aked whether the length is at most $2n+O(1)$. This is true, as a consequence of result of Tao [Ta25] below.\n\nLet the maximal length of such a curve be denoted by $f(n)$.\nThe length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular that $f(n)=2n+O(1)$.\nDolzhenko [Do61] proved $f(n) \\leq 4\\pi n$, but few were aware of this work.\nPommerenke [Po61] proved $f(n)\\ll n^2$.\nBorwein [Bo95] proved $f(n)\\ll n$ (Borwein was unaware of Dolzhenko's earlier work). The prize of \\$250 is reported by Borwein [Bo95].\nEremenko and Hayman [ErHa99] proved the full conjecture when $n=2$, and $f(n)\\leq 9.173n$ for all $n$.\nDanchenko [Da07] proved $f(n)\\leq 2\\pi n$.\nFryntov and Nazarov [FrNa09] proved that $z^n-1$ is a local maximiser, and solved this problem asymptotically, proving that\\[f(n)\\leq 2n+O(n^{7/8}).\\]\n Tao [Ta25] has proved that $p(z)=z^n-1$ is the unique (up to rotation and translation) maximiser for all sufficiently large $n$.\nErdős, Herzog, and Piranian [EHP58] also ask whether the length is at least $2\\pi$ if $\\{ z: \\lvert f(z)\\rvert<1\\}$ is connected (which $z^n$ shows is the best possible). This was proved by Pommerenke [Po59].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#114: [EHP58,p.142][Er61,p.247][Ha74][Er82e][Er90][Er97f][Va99,2.35]" }, { "number": 117, "url": "https://www.erdosproblems.com/117", "status": "open", "prize": "no", "tags": [ "group theory" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.Estimate $h(n)$ as well as possible.", "commentary": "Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n0$ such that for infinitely many $n$ we have $M_n > n^c$?Is it true that there exists $c>0$ such that, for all large $n$,\\[\\sum_{k\\leq n}M_k > n^{1+c}?\\]", "commentary": "This is Problem 4.1 in [Ha74] where it is attributed to Erdős.\n\nThe weaker conjecture that $\\limsup M_n=\\infty$ was proved by Wagner [Wa80], who show that there is some $c>0$ with $M_n>(\\log n)^c$ infinitely often.\n\nThe second question was answered by Beck [Be91], who proved that there exists some $c>0$ such that\\[\\max_{n\\leq N} M_n > N^c.\\]Erdős (e.g. see [Ha74]) gave a construction of a sequence with $M_n\\leq n+1$ for all $n$. Linden [Li77] improved this to give a sequence with $M_n\\ll n^{1-c}$ for some $c>0$.\n\nThe third question seems to remain open.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#119: [Er57][Er61][Er64b][Ha74][Er82e][Er90][Er97f][Va99,2.38]" }, { "number": 120, "url": "https://www.erdosproblems.com/120", "status": "open", "prize": "$100", "tags": [ "combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq\\mathbb{R}$ be an infinite set. Must there be a set $E\\subset \\mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\\in\\mathbb{R}$ and $a\\neq 0$?", "commentary": "The Erdős similarity problem. \n\nThis is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\\{a_1>a_2>\\cdots\\}$ is a countable strictly monotone sequence which converges to $0$. \n\nSteinhaus [St20] has proved this is false whenever $A$ is a finite set. \n\nThis conjecture is known in many special cases (but, for example, it is open when $A=\\{1,1/2,1/4,\\ldots\\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen [JLM24].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#120: [Er74b][Er81b,p.29][Er83d][Er90][Er97f][Va99,2.46]" }, { "number": 122, "url": "https://www.erdosproblems.com/122", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", "commentary": "In [Er97] Erdős is only considering number theoretic functions which grow 'slowly' (i.e. slower than $(\\log n)^{1-c}$ for some $c>0$). \n\nThis is considered by Erdős, Pomerance, and Sárközy [EPS97], who prove in particular that if $\\omega(n)$ counts the number of distinct prime divisors of $n$, then for all large $x$ there are intervals $I,J\\subset [1,x]$ of width\\[\\lvert I\\rvert\\asymp \\left(\\frac{\\log x}{\\log\\log x}\\right)^{1/2}\\]and\\[\\lvert J\\rvert \\asymp (\\log\\log x)^{1/2}\\]such that if $n\\in I$ then $n+\\omega(n)\\in J$. (Note the normal order of $\\omega$ is $\\log\\log x$, and hence $(\\log\\log n)^{1/2}/\\omega(n)\\to 0$ for almost all $n$.) \n\nIn [Er97] and [Er97e] Erdős reports that he, Pomerance, and Sárkzözy can prove the more general claim above for $f$ being $\\tau(n)$, the divisor function, or $\\omega(n)$, and states it 'probably fails' for $\\phi(n)$ or $\\sigma(n)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#122: [Er97,p.155][Er97e,p.533][EPS97]" }, { "number": 123, "url": "https://www.erdosproblems.com/123", "status": "open", "prize": "$250", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $a,b,c\\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\\geq 0$), none of which divide any other?", "commentary": "A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.\n\nAs a partial record of progress so far, the sequence $\\{a^kb^lc^m\\}$ is known to be $d$-complete when:\n$a=3$, $b=5$, $c=7$ (Erdős and Lewin [ErLe96]).\n$a=2$, $b=5$, $c\\in \\{7,11,13,17,19\\}$ (Erdős and Lewin [ErLe96]).\n$a=2$, $b=5$, $c\\in \\{9,21,23,27,29,31\\}$ - more generally, $a=2$, $b=5$, and any $c>6$ with $(c,10)=1$ such that there exists $N$ where every integer in $(N,25cN)$ is the sum of distinct elements of $\\{2^k3^lc^m\\}$, none of which divide any other (Ma and Chen [MaCh16]).\n $a=2$, $b=5$, $3\\leq c\\leq 87$ with $(c,10)=1$, or $a=2$, $b=7$, $3\\leq c\\leq 33$ with $(c,14)=1$, or $a=3$, $b=5$, $2\\leq c\\leq 14$ with $(c,15)=1$ (Chen and Yu [ChYu23b]).\nIn [Er92b] Erdős makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\\cdots 0$, of an infinite set of $d_i$ for which every sufficiently large integer can be written as a finite sum of the shape $\\sum_i c_ia_i$ where $c_i\\in \\{0,1\\}$ and $a_i\\in P(d_i,0)$ and yet $\\sum_{i}\\frac{1}{d_i-1}<\\epsilon$.\n\n\nSee also [125].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 December 2025. View history", "references": "#124: [BEGL96][Er97,p.156][Er97e,p.533]" }, { "number": 126, "url": "https://www.erdosproblems.com/126", "status": "open", "prize": "$250", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $\\lvert A\\rvert=n$ then $\\prod_{a\\neq b\\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\\log n\\to\\infty$?", "commentary": "Investigated by Erdős and Turán [ErTu34] (prompted by a question of Lázár and Grünwald) in their first joint paper, where they proved that\\[\\log n \\ll f(n) \\ll n/\\log n\\](the upper bound is trivial, taking $A=\\{1,\\ldots,n\\}$). Erdős says that $f(n)=o(n/\\log n)$ has never been proved, but perhaps never seriously attacked.\n\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#126: [ErTu34][Er95c][Er97][Er97e]" }, { "number": 128, "url": "https://www.erdosproblems.com/128", "status": "falsifiable", "prize": "$250", "tags": [ "graph theory" ], "oeis": [], "formalized": "yes", "statement": "Let $G$ be a graph with $n$ vertices such that every induced subgraph on $\\geq \\lfloor n/2\\rfloor$ vertices has more than $n^2/50$ edges. Must $G$ contain a triangle?", "commentary": "A problem of Erdős and Rousseau. The constant $50$ would be best possible as witnessed by a blow-up of $C_5$ or the Petersen graph. \n\nErdős, Faudree, Rousseau, and Schelp [EFRS94] proved that this is true with $50$ replaced by $16$. More generally, they prove that, for any $0<\\alpha<1$, if every set of $\\geq \\alpha n$ vertices contains $>\\alpha^3n^2/2$ edges then $G$ contains a triangle.\n\nKrivelevich [Kr95] has proved this with $n/2$ replaced by $3n/5$ (and $50$ replaced by $25$).\n\nKeevash and Sudakov [KeSu06] have proved this under the additional assumption that either $G$ has at most $n^2/12$ edges, or that $G$ has at least $n^2/5$ edges. Norin and Yepremyan [NoYe15] proved that this is true if $G$ has at least $(1/5-c)n^2$ edges, for some constant $c>0$.\n\nRazborov [Ra22] proved this is true if $\\frac{1}{50}$ is replaced by $\\frac{27}{1024}$.\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#128: [Er93,p.344][ErRo93][Er97b]" }, { "number": 129, "url": "https://www.erdosproblems.com/129", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\\[R(n;3,r) < C^{\\sqrt{n}}.\\]", "commentary": "Conjectured by Erdős and Gyárfás, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erdős thought it likely that for all $r,k\\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that\\[ C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}.\\]Antonio Girao has pointed out that this problem as written is easily disproved, and indeed $R(n;3,2) \\geq C^{n}$:\n\nThe obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of $K_N$ such every set on $n$ vertices contains a red triangle and a blue triangle (using that every set of $n$ vertices contains $\\gg n^2$ edge-disjoint triangles), provided $N \\leq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \\geq C^{n}$, contradicting the conjecture. \n\n\nPerhaps Erdős had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like $C^{\\sqrt{n}}$ as Erdős and Gyárfás claim to have achieved via the probabilistic method.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#129: [Er97b]" }, { "number": 130, "url": "https://www.erdosproblems.com/130", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart. How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?", "commentary": "Asked by Andrásfai and Erdős. Erdős [Er97b] also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erdős [AnEr45].\n\nSee also [213].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#130: [Er97b]" }, { "number": 131, "url": "https://www.erdosproblems.com/131", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A068063" ], "formalized": "no", "statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any distinct elements of $A\\backslash\\{a\\}$. Estimate $F(N)$. In particular, is it true that\\[F(N) > N^{1/2-o(1)}?\\]", "commentary": "This was studied by Erdős, Lev, Rauzy, Sándor, and Sárközy [ELRSS99], where they call such a property 'non-dividing', and prove the explicit bound\\[F(N)<3N^{1/2}+1.\\]In [Er97b] Erdős credits Csaba with a construction that proves $F(N) \\gg N^{1/5}$. Such a construction was also given in [ELRSS99], where it is linked to the problem of non-averaging sets (see [186]).\n\nIndeed, every such set is non-averaging, and hence the result of Pham and Zakharov [PhZa24] implies\\[F(N) \\leq N^{1/4+o(1)}.\\]This shows the answer to the original question is no, but the general question of the correct growth of $F(N)$ remains open.\n\nIn [Er75b] Erdős writes that he originally thought $F(N) <(\\log N)^{O(1)}$, but that Straus proved that\\[F(N) > \\exp((\\sqrt{\\tfrac{2}{\\log 2}}+o(1))\\sqrt{\\log N}).\\]See also [13].\n\nThis is discussed in problem C16 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#131: [Er75b,p.309][Er97b,p.230][ELRSS99,p.129]" }, { "number": 132, "url": "https://www.erdosproblems.com/132", "status": "open", "prize": "$100", "tags": [ "distances" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\\to \\infty$ as $n\\to \\infty$?", "commentary": "Asked by Erdős and Pach. Hopf and Pannowitz [HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.\n\nIt may be true that there are at least $n^{1-o(1)}$ many such distances. In [Er97e] Erdős offers \\$100 for 'any nontrivial result'.\n\nErdős [Er84c] believed that for $n\\geq 5$ there must always exist at least two such distances. This is false for $n=4$, as witnessed by two equilateral triangles of the same side-length glued together. Erdős and Fishburn [ErFi95] proved this is true for $n=5$ and $n=6$.\n\nClemen, Dumitrescu, and Liu [CDL25] have proved that there always at least two such distances if $A$ is in convex position (that is, no point lies inside the convex hull of the others). They also prove it is true if the set $A$ is 'not too convex', in a specific technical sense.\n\nSee also [223], [756], and [957].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#132: [Er84c][ErPa90][ErFi95][Er97b][Er97e]" }, { "number": 137, "url": "https://www.erdosproblems.com/137", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "We say that $N$ is powerful if whenever $p\\mid N$ we also have $p^2\\mid N$. Let $k\\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?", "commentary": "Conjectured by Erdős and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erdős and Selfridge [ErSe75] proved that the product of $k\\geq 3$ consecutive positive integers can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.\n\nIn [Er82c] he further conjectures that, if $k$ is fixed and $n$ is sufficiently large, then, for all $m$, there must be at least $k$ distinct primes $p$ such that\\[p\\mid m(m+1)\\cdots (m+n)\\]and yet $p^2$ does not divide the right-hand side.\n\nSee also [364].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 January 2026. View history", "references": "#137: [ErGr80][Er82c,p.28][Er97c]" }, { "number": 138, "url": "https://www.erdosproblems.com/138", "status": "open", "prize": "$500", "tags": [ "additive combinatorics" ], "oeis": [ "A005346" ], "formalized": "yes", "statement": "Let the van der Waerden number $W(k)$ be such that whenever $N\\geq W(k)$ and $\\{1,\\ldots,N\\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\\to \\infty$.", "commentary": "When $p$ is prime Berlekamp [Be68] has proved $W(p+1)\\geq p2^p$. Gowers [Go01] has proved\\[W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}}.\\]The best general lower bound is $W(k)\\gg 2^k$, due to Kozik and Shabanov [KoSh16].\n\nIn [Er81] Erdős further asks whether $W(k+1)/W(k)\\to \\infty$, or $W(k+1)-W(k)\\to \\infty$. DeepMind has proved (see the comments) that\\[W(k+1)\\geq W(k)+k,\\]answering the second question.\n\nIn [Er80] Erdős asks whether $W(k)/2^k\\to \\infty$, and offers \\$500 for a proof or disproof of $W(k)^{1/k}\\to \\infty$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#138: [Er57][Er61][Er73][Er74b][Er75b][Er77c][ErGr79][Er80,p.90][ErGr80][Er81][Er97c]" }, { "number": 141, "url": "https://www.erdosproblems.com/141", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "primes", "arithmetic progressions" ], "oeis": [ "A006560" ], "formalized": "yes", "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?", "commentary": "Green and Tao [GrTa08] have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erdős called this conjecture 'completely hopeless at present'.\n\nThe existence of such progressions for small $k$ has been verified for $k\\leq 10$, see the Wikipedia page. It is open, even for $k=3$, whether there are infinitely many such progressions.\n\nSee also [219].\n\nThis is discussed in problem A6 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#141: [Er75b][Er83][Er97c]" }, { "number": 142, "url": "https://www.erdosproblems.com/142", "status": "open", "prize": "$10000", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A003002", "A003003", "A003004", "A003005" ], "formalized": "yes", "statement": "Let $r_k(N)$ be the largest possible size of a subset of $\\{1,\\ldots,N\\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.", "commentary": "Erdős remarked this is 'probably unattackable at present'. In [Er97c] Erdős offered \\$1000, but given that he elsewhere offered \\$5000 just for (essentially) showing that $r_k(N)=o_k(N/\\log N)$ (see [3]), that value seems odd. In [Er81] he offers \\$10000, stating it is 'probably enormously difficult'.\n\nThe best known upper bounds for $r_k(N)$ are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.\n\nIn [Er80] and [Va99] he asks (much more reasonably) for the order of magnitude of $r_k(N)$. In [Er80] he remarks that we do not even know whether $r_k(n)/r_{k+1}(n)\\to 0$ for any $k\\geq 3$.\n \nSee also [3] and [139].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 April 2026. View history", "references": "#142: [Er80,p.92][Er81,p.4][Er97c][Va99,1.27]" }, { "number": 143, "url": "https://www.erdosproblems.com/143", "status": "open", "prize": "$500", "tags": [ "primitive sets" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset (1,\\infty)$ be a countably infinite set such that for all $x\\neq y\\in A$ and integers $k\\geq 1$ we have\\[ \\lvert kx -y\\rvert \\geq 1.\\]Does this imply that $A$ is sparse? In particular, does this imply that\\[\\sum_{x\\in A}\\frac{1}{x\\log x}<\\infty\\]or\\[\\sum_{\\substack{x 0$ is some absolute constant and $c_0=1.26408\\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 September 2025. View history", "references": "#148: [ErGr80,p.32]" }, { "number": 149, "url": "https://www.erdosproblems.com/149", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "The strong chromatic index of a graph $G$, denoted by $\\mathrm{sq}(G)$, is the minimum $k$ such that the edges of $G$ can be partitioned into $k$ sets of 'strongly independent' edges, that is, such that the subgraph of $G$ induced by each set is the union of vertex-disjoint edges.Is it true that, for any graph $G$ with maximum degree $\\Delta$,\\[\\mathrm{sq}(G)\\leq\\frac{5}{4}\\Delta^2?\\]", "commentary": "Asked by Erdős and Nešetřil in 1985 (see [FGST89]). This is equivalent to asking whether the chromatic number of the square of the line graph $L(G)^2$ is at most $\\frac{5}{4}\\Delta^2$. This bound would be the best possible, as witnessed by a blowup of $C_5$, although improvements may be possible for odd $\\Delta$.\n\nIt is easy to see that $\\mathrm{sq}(G)\\leq 5$ when $\\Delta\\leq 2$: in general, the trivial bound of\\[\\mathrm{sq}(G)\\leq 2\\Delta^2-2\\Delta+1\\]follows from the observations that the number of neighbours of any edge of $G$ is $\\leq 2\\Delta^2-2\\Delta$.\n\nThis was improved to $\\mathrm{sq}(G)\\leq 1.998\\Delta^2$ (for $\\Delta$ sufficiently large) by Molloy and Reed [MoRe97]. This was improved to $1.93\\Delta^2$ by Bruhn and Joos [BrJo18] and to $1.835\\Delta^2$ by Bonamy, Perrett, and Postle [BPP22]. The best bound currently available is\\[1.772\\Delta^2,\\]proved by Hurley, de Joannis de Verclos, and Kang [HJK22]. Mahdian has, in their Masters' thesis, proved an upper bound of $(2+o(1))\\frac{\\Delta^2}{\\log \\Delta}$ under the additional assumption that $G$ is $C_4$-free.\n\nAndersen [92] and Horák, He, and Trotter [HHT93] have independently shown that $\\mathrm{sq}(G)\\leq 10$ when $\\Delta\\leq 3$. (This is best possible, as shown e.g. by a $C_8$ with all four diagonals). Huang, Santana, and Yu [HSY18] proved that $\\mathrm{sq}(G)\\leq 21$ when $\\Delta\\leq 4$.\n\nErdős and Nešetřil also asked the easier problem of whether $G$ containing at least $\\tfrac{5}{4}\\Delta^2$ many edges implies $G$ containing two strongly independent edges. This was proved by Chung, Gyárfás, Tuza, and Trotter [CGTT90].\n\nIt is still open even whether the clique number of $L(G)^2$ at most $\\frac{5}{4}\\Delta^2$. Let $\\omega=\\omega(L(G)^2)$ be this clique number. Śleszyńska-Nowak [Sl16] proved $\\omega \\leq \\frac{3}{2}\\Delta^2$. Faron and Postle [FaPo19] proved $\\omega\\leq \\frac{4}{3}\\Delta^2$. Cames van Batenburg, Kang, and Pirot [CKP20] have proved $\\omega\\leq \\frac{5}{4}\\Delta^2$ under the additional assumption that $G$ is triangle-free (and $\\omega\\leq \\Delta^2$ if $G$ is $C_5$-free).\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#149: [Er88,p.81]" }, { "number": 151, "url": "https://www.erdosproblems.com/151", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).Let $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.If $G$ is a graph on $n$ vertices then is\\[\\tau(G)\\leq n-H(n)?\\]", "commentary": "It is easy to see that $\\tau(G) \\leq n-\\sqrt{n}$. Note also that if $G$ is triangle-free then trivially $\\tau(G)\\leq n-H(n)$.\n\nThis is listed in [Er88] as a problem of Erdős and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. There Erdős remarked that this conjecture is 'perhaps completely wrongheaded'. \n\nIt later appeared as Problem 1 in [EGT92].\n\nThe general behaviour of $\\tau(G)$ is the subject of [610].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#151: [Er88,p.82][EGT92,p.280]" }, { "number": 153, "url": "https://www.erdosproblems.com/153", "status": "open", "prize": "no", "tags": [ "sidon sets" ], "oeis": [], "formalized": "yes", "statement": "Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots0$ such that$$R(C_4,K_n) \\ll n^{2-c}.$$", "commentary": "The prize of \\$100 is offered in [Er78] for a proof or disproof. The current bounds are\\[ \\frac{n^{3/2}}{(\\log n)^{3/2}}\\ll R(C_4,K_n)\\ll \\frac{n^2}{(\\log n)^2}.\\]The upper bound is due to Szemerédi (mentioned in [EFRS78]), and the lower bound is due to Spencer [Sp77].\n\nThis problem is #17 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#159: [Er78,p.34][Er81][Er84d]" }, { "number": 160, "url": "https://www.erdosproblems.com/160", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [], "formalized": "yes", "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.", "commentary": "Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows\\[h(N) \\ll N^{2/3}.\\]In the comments of this site Hunter improves this to\\[h(N) \\ll N^{\\frac{\\log 3}{\\log 22}+o(1)}\\](note $\\frac{\\log 3}{\\log 22}\\approx 0.355$).\n\nThe observation of Zach Hunter in that question coupled with recent progress on the size of subsets without three-term arithmetic progression (see [BlSi23] which improves slightly on the bounds due to Kelley and Meka [KeMe23]) imply that\\[h(N) \\gg \\exp(c(\\log N)^{1/9})\\]for some $c>0$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#160: [Er89]" }, { "number": 161, "url": "https://www.erdosproblems.com/161", "status": "open", "prize": "$500", "tags": [ "combinatorics", "ramsey theory", "discrepancy" ], "oeis": [], "formalized": "no", "statement": "Let $\\alpha\\in[0,1/2)$ and $n,t\\geq 1$. Let $F^{(t)}(n,\\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\\subseteq [n]$ with $\\lvert X\\rvert \\geq m$ then there are at least $\\alpha \\binom{\\lvert X\\rvert}{t}$ many $t$-subsets of $X$ of each colour. For fixed $n,t$ as we change $\\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\\alpha)$ increase continuously or are there jumps? Only one jump?", "commentary": "For $\\alpha=0$ this is the usual Ramsey function.\n\nA conjecture of Erdős, Hajnal, and Rado (see [562]) implies that\\[ F^{(t)}(n,0)\\asymp \\log_{t-1} n\\]and results of Erdős and Spencer imply that\\[F^{(t)}(n,\\alpha) \\gg_\\alpha (\\log n)^{\\frac{1}{t-1}}\\]for all $\\alpha>0$, and a similar upper bound holds for $\\alpha$ close to $1/2$. \n\nErdős said in [Er90b]: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at $0$. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for $t>3$. I know nothing and offer \\$500 to anybody who can clear up this mystery.' \n\nConlon, Fox, and Sudakov [CFS11] have proved that, for any fixed $\\alpha>0$,\\[F^{(3)}(n,\\alpha) \\ll_\\alpha \\sqrt{\\log n}.\\]Coupled with the lower bound above, this implies that there is only one jump for fixed $\\alpha$ when $t=3$, at $\\alpha=0$.\n\nFor all $\\alpha>0$ it is known that\\[F^{(t)}(n,\\alpha)\\gg_t (\\log n)^{c_\\alpha}.\\]See also [563] for more on the case $t=2$.\n\nThis problem is #40 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 January 2026. View history", "references": "#161: [Er90b,p.21]" }, { "number": 162, "url": "https://www.erdosproblems.com/162", "status": "open", "prize": "no", "tags": [ "combinatorics", "ramsey theory", "discrepancy" ], "oeis": [], "formalized": "no", "statement": "Let $\\alpha>0$ and $n\\geq 1$. Let $F(n,\\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\\alpha\\binom{\\lvert H\\rvert}{2}$ many edges of each colour.Prove that for every fixed $0\\leq \\alpha \\leq 1/2$, as $n\\to\\infty$,\\[F(n,\\alpha)\\sim c_\\alpha \\log n\\]for some constant $c_\\alpha$.", "commentary": "It is easy to show with the probabilistic method that there exist $c_1(\\alpha),c_2(\\alpha)$ such that\\[c_1(\\alpha)\\log n < F(n,\\alpha) < c_2(\\alpha)\\log n.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 December 2025. View history", "references": "#162: [Er90b,p.21]" }, { "number": 165, "url": "https://www.erdosproblems.com/165", "status": "open", "prize": "$250", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A000791" ], "formalized": "no", "statement": "Give an asymptotic formula for $R(3,k)$.", "commentary": "It is known that there exists some constant $c>0$ such that for large $k$\\[(c+o(1))\\frac{k^2}{\\log k}\\leq R(3,k) \\leq (1+o(1))\\frac{k^2}{\\log k}.\\]The lower bound is due to Kim [Ki95], the upper bound is due to Shearer [Sh83], improving an earlier bound of Ajtai, Komlós, and Szemerédi [AKS80]. \n\nThe value of $c$ in the lower bound has seen a number of improvements. Kim's original proof gave $c\\geq 1/162$. The bound $c\\geq 1/4$ was proved independently by Bohman and Keevash [BoKe21] and Pontiveros, Griffiths and Morris [PGM20]. The latter collection of authors conjecture that this lower bound is the true order of magnitude.\n\nThis was, however, improved by Campos, Jenssen, Michelen, and Sahasrabudhe [CJMS25] to $c\\geq 1/3$, and further by Hefty, Horn, King, and Pfender [HHKP25] to $c\\geq 1/2$. Both of these papers conjecture that $c=1/2$ is the correct asymptotic.\n\nSee also [544], and [986] for the general case. See [1013] for a related function.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#165: [Er61][Er71][Er78,p.34][Er90b][Er93,p.339][Er97c]" }, { "number": 167, "url": "https://www.erdosproblems.com/167", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?", "commentary": "A problem of Tuza. It is trivial that $G$ can be made triangle-free after removing at most $3k$ edges. The examples of $K_4$ and $K_5$ show that $2k$ would be best possible.\n\nThe trivial bound of $\\leq 3k$ was improved to $\\leq (3-\\frac{3}{23}+o(1))k$ by Haxell [Ha99].\n\nKahn and Park [KaPa22] have proved this is true for random graphs.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 13 October 2025. View history", "references": "#167: [Er88]" }, { "number": 168, "url": "https://www.erdosproblems.com/168", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [ "A004059", "A057561", "A094708", "A386439" ], "formalized": "yes", "statement": "Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is\\[ \\lim_{N\\to \\infty}\\frac{F(N)}{N}?\\]Is this limit irrational?", "commentary": "This limit was proved to exist by Graham, Spencer, and Witsenhausen [GSW77], who showed it is equal to\\[\\frac{1}{3}\\sum_{k\\in K}\\frac{1}{d_k},\\]where $d_1f(k-1)$, where $f$ counts the largest subset of $\\{d_1,\\ldots,d_k\\}$ that avoids $\\{n,2n,3n\\}$.\n\nSimilar questions can be asked for the density or upper density of infinite sets without such configurations.\n\nThe limit can be estimated by elementary arguments (see the comments). Eberhard has used the formula of [GSW77] mentioned above to calculate the value of the limit as\\[0.800965\\cdots.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#168: [ErGr79][ErGr80]" }, { "number": 169, "url": "https://www.erdosproblems.com/169", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A005346" ], "formalized": "no", "statement": "Let $k\\geq 3$ and $f(k)$ be the supremum of $\\sum_{n\\in A}\\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$. Is\\[\\lim_{k\\to \\infty}\\frac{f(k)}{\\log W(k)}=\\infty\\]where $W(k)$ is the van der Waerden number?", "commentary": "Berlekamp [Be68] proved $f(k) \\geq \\frac{\\log 2}{2}k$. Gerver [Ge77] proved\\[f(k) \\geq (1-o(1))k\\log k.\\]It is trivial that\\[\\frac{f(k)}{\\log W(k)}\\geq \\frac{1}{2},\\]but improving the right-hand side to any constant $>1/2$ is open.\n\nGerver also proved (see the comments for an alternative argument of Tao) that [3] is equivalent to $f(k)$ being finite for all $k$.\n\nThe current record for $f(3)$ is $f(3)\\geq 3.00849$, due to Wróblewski [Wr84]. Walker [Wa25] proved $f(4)\\geq 4.43975$.\n\nWalker [Wa25] has shown that it suffices to consider Kempner sets (that is, sets of integers defined as all those whose base $b$ digits are contained in some $S\\subset \\{0,\\ldots,b-1\\}$ for fixed $b$ and $S$), in the sense that for any $k\\geq 3$ and $\\epsilon>0$ there is a Kempner set $A$ lacking $k$-term arithmetic progressions such that\\[\\sum_{n\\in A}\\frac{1}{n}\\geq f(k)-\\epsilon.\\]In [Er80] Erdős asks whether for every $\\epsilon>0$ and $k\\geq 3$ if $A$ is a set of integers without a $k$-term arithmetic progression such that $\\min(A)$ is sufficiently large in terms of $\\epsilon$ and $k$ then $\\sum_{n\\in A}\\frac{1}{n}<\\epsilon$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 April 2026. View history", "references": "#169: [Er77c][ErGr79][Er80,p.92][ErGr80]" }, { "number": 170, "url": "https://www.erdosproblems.com/170", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [ "A046693" ], "formalized": "yes", "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of\\[\\lim_{N\\to \\infty}\\frac{F(N)}{N^{1/2}}.\\]", "commentary": "The Sparse Ruler problem. Rédei asked whether this limit exists, which was proved by Erdős and Gál [ErGa48]. Bounds on the limit were improved by Leech [Le56]. The limit is known to be in the interval $[1.56,\\sqrt{3}]$. The lower bound is due to Leech [Le56], the upper bound is due to Wichmann [Wi63]. Computational evidence by Pegg [Pe20] suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\\subset \\{0,1,\\ldots,N\\}$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#170: [ErGa48]" }, { "number": 172, "url": "https://www.erdosproblems.com/172", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?", "commentary": "First asked by Hindman. Hindman [Hi80] has proved this is false (with 7 colours) if we ask for an infinite $A$. In [Er77c] Erdős asks about the case for an infinite $A$ with just $2$ colours (see [1198]).\n\nMoreira [Mo17] has proved that in any finite colouring of $\\mathbb{N}$ there exist $x,y$ such that $\\{x,x+y,xy\\}$ are all the same colour.\n\nAlweiss [Al23] has proved that in any finite colouring of $\\mathbb{Q}\\backslash \\{0\\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok [BoSa22] had proved this earlier for the first non-trivial case of $\\lvert A\\rvert=2$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#172: [Er77c][ErGr79][ErGr80]" }, { "number": 173, "url": "https://www.erdosproblems.com/173", "status": "open", "prize": "no", "tags": [ "geometry", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "In any $2$-colouring of $\\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.", "commentary": "For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader [Sh76] has proved this is true if we just consider a single right-angled triangle.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 October 2025. View history", "references": "#173: [Er75f,p.108][ErGr79][ErGr80][Er83c]" }, { "number": 174, "url": "https://www.erdosproblems.com/174", "status": "open", "prize": "no", "tags": [ "geometry", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "A finite set $A\\subset \\mathbb{R}^n$ is called Ramsey if, for any $k\\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\\mathbb{R}^n$.", "commentary": "Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS73] proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters [LRW12] have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some higher-dimensional set on which rotations act transitively.\n\nSets known to be Ramsey include vertices of $k$-dimensional rectangles [EGMRSS73], non-degenerate simplices [FrRo90], trapezoids [Kr92], and regular polygons/polyhedra [Kr91]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 October 2025. View history", "references": "#174: [Er75f,p.108][ErGr79][ErGr80][Er83c]" }, { "number": 176, "url": "https://www.erdosproblems.com/176", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions", "discrepancy" ], "oeis": [], "formalized": "no", "statement": "Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that\\[ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell.\\]Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that\\[N(k,ck)\\leq C^k?\\]What about\\[N(k,2)\\leq C^k\\]or\\[N(k,\\sqrt{k})\\leq C^k?\\]", "commentary": "When $\\ell=k$ this is the van der Waerden number $W(k)$ (see [138]). Spencer [Sp73] has proved that if $k=2^tm$ with $m$ odd then\\[N(k,1)=2^t(k-1)+1.\\]Erdős and Graham write that 'no decent bound' is known even for $N(k,2)$.\n\nErdős [Er63d] proved that, for every $c>0$,\\[N(k,ck)> (1+\\alpha_c)^k\\]where $\\alpha_c\\to 0$ as $c\\to 0$ and $\\alpha_c\\to \\sqrt{2}-1$ as $c\\to 1$. \n\nHunter in the comment section observes that the local lemma implies an improved bound of\\[N(k,ck) \\gg \\frac{2^k}{k^{O(1)}\\sum_{i>\\frac{1+c}{2}k}\\binom{k}{i}},\\]so that in particular as $c\\to 1$ we have, for all large $k$, $N(k,ck)\\geq (2-o(1))^k$ (where the $o(1)$ term $\\to 0$ as $c\\to 1$). \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 April 2026. View history", "references": "#176: [Er65b][Er73][Er74b][Er75b][ErGr79][Er80,p.91][ErGr80,p.15]" }, { "number": 177, "url": "https://www.erdosproblems.com/177", "status": "open", "prize": "no", "tags": [ "discrepancy", "arithmetic progressions" ], "oeis": [], "formalized": "no", "statement": "Find the smallest $h(d)$ such that the following holds. There exists a function $f:\\mathbb{N}\\to\\{-1,1\\}$ such that, for every $d\\geq 1$,\\[\\max_{P_d}\\left\\lvert \\sum_{n\\in P_d}f(n)\\right\\rvert\\leq h(d),\\]where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.", "commentary": "Cantor, Erdős, Schreiber, and Straus [Er66] proved that $h(d)\\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\\to \\infty$. Beck [Be17] has shown that $h(d) \\leq d^{8+\\epsilon}$ is possible for every $\\epsilon>0$. Roth's famous discrepancy lower bound [Ro64] implies that $h(d)\\gg d^{1/2}$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#177: [Er73][ErGr79][ErGr80]" }, { "number": 180, "url": "https://www.erdosproblems.com/180", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$. Is it true that, for every $\\mathcal{F}$, there exists $G\\in\\mathcal{F}$ such that\\[\\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})?\\]", "commentary": "A problem of Erdős and Simonovits.\n\nThis is trivially true if $\\mathcal{F}$ does not contain any bipartite graphs, since by the Erdős-Stone theorem if $H\\in\\mathcal{F}$ has minimal chromatic number $r\\geq 2$ then\\[\\mathrm{ex}(n;H)=\\mathrm{ex}(n;\\mathcal{F})=\\left(\\frac{r-2}{r-1}+o(1)\\right)\\binom{n}{2}.\\]Erdős and Simonovits observe that this is false for infinite families $\\mathcal{F}$, e.g. the family of all cycles.\n\n\nHunter has provided the following 'folklore counterexample': if $\\mathcal{F}=\\{H_1,H_2\\}$ where $H_1$ is a star and $H_2$ is a matching, both with at least two edges, then $\\mathrm{ex}(n;\\mathcal{F})\\ll 1$, but $\\mathrm{ex}(n;H_i)\\asymp n$ for $1\\leq i\\leq 2$. This conjecture may still hold for all other $\\mathcal{F}$. \n\nSee also [575].\n\nThis problem is #47 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#180: [ErSi82]" }, { "number": 181, "url": "https://www.erdosproblems.com/181", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that\\[R(Q_n) \\ll 2^n.\\]", "commentary": "Conjectured by Burr and Erdős, althouhg in [Er93] Erdős says the behaviour of $R(Q_n)$ was considered by himself and Sós, who could not decide whether $R(Q_n)/2^n\\to \\infty$ or not.\n\nThe trivial bound is\\[R(Q_n) \\leq R(K_{2^n})\\leq C^{2^n}\\]for some constant $C>1$. This was improved a number of times; the current best bound due to Tikhomirov [Ti22] is\\[R(Q_n)\\ll 2^{(2-c)n}\\]for some small constant $c>0$. (In fact $c\\approx 0.03656$ is permissible.)\n\nThis problem is #20 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#181: [BuEr75][Er93,p.346]" }, { "number": 183, "url": "https://www.erdosproblems.com/183", "status": "open", "prize": "$250", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A003323" ], "formalized": "no", "statement": "Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine\\[\\lim_{k\\to \\infty}R(3;k)^{1/k}.\\]", "commentary": "Erdős offers \\$100 for showing that this limit is finite. An easy pigeonhole argument shows that\\[R(3;k)\\leq 2+k(R(3;k-1)-1),\\]from which $R(3;k)\\leq \\lceil e k!\\rceil$ immediately follows. The best-known upper bound is\\[R(3;k)\\leq (e-\\tfrac{1}{6}) k!+1,\\]due to Xu, Xie, and Chen [XXC02] (improving previous bounds of Wan [Wa97] and Whitehead [Wh73]). See Eliahou [El19] for more on the upper bound. These arise from the same type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is\\[R(3,k)\\geq (380)^{k/5}-O(1),\\]due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik [ACPPRT21] (improving previous bounds of Exoo [Ex94] and Fredricksen and Sweet [FrSw00]). Note that $380^{1/5}\\approx 3.2806$.\n\nSee also [483].\n\nThis problem is #21 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#183: [Er61]" }, { "number": 184, "url": "https://www.erdosproblems.com/184", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "yes", "statement": "Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.", "commentary": "Conjectured by Erdős and Gallai, who proved that $O(n\\log n)$ many cycles and edges suffices (see Section 5 of [EGP66]). \n\nThe graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$. In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint.\n\nThe best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\\log^*n)$ many cycles and edges suffice, where $\\log^*$ is the iterated logarithm function.\n\nConlon, Fox, and Sudakov [CFS14] proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\\epsilon n$, for any $\\epsilon>0$.\n\nSee also [583] for an analogous problem decomposing into paths, and [1017] for decomposing into complete graphs.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#184: [EGP66][Er71][Er76][Er81][Er83b]" }, { "number": 187, "url": "https://www.erdosproblems.com/187", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.", "commentary": "Originally asked by Cohen. \n\nErdős observed that colouring according to whether $\\{ \\sqrt{2}n\\}<1/2$ or not implies $f(d) \\ll d$ (using the fact that $\\|\\sqrt{2}q\\| \\gg 1/q$ for all $q$, where $\\|x\\|$ is the distance to the nearest integer). In [Er80] he states that Petruska and Szemerédi have proved that $f(d) \\ll d^{1/2}$, and expected $f(d)\\leq d^{o(1)}$.\n\nBeck [Be80] has improved this using the probabilistic method, constructing a colouring that shows $f(d)\\leq (1+o(1))\\log_2 d$. Van der Waerden's theorem implies $f(d)\\to \\infty$ is necessary.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 April 2026. View history", "references": "#187: [Er73][ErGr79][Er80,p.93][ErGr80,p.17]" }, { "number": 188, "url": "https://www.erdosproblems.com/188", "status": "open", "prize": "no", "tags": [ "geometry", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?", "commentary": "Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] proved $k\\geq 5$. Tsaturian [Ts17] improved this to $k\\geq 6$. Erdős and Graham claim that $k\\leq 10000000$ ('more or less'), but give no proof.\n\nErdős and Graham asked this with just any $k$-term arithmetic progression in blue (not necessarily with distance $1$), but Alon has pointed out that in fact no such $k$ exists: in any red/blue colouring of the integer points on a line either there are two red points distance $1$ apart, or else the set of blue points and the same set shifted by $1$ cover all integers, and hence by van der Waerden's theorem there are arbitrarily long blue arithmetic progressions.\n\nIt seems most likely, from context, that Erdős and Graham intended to restrict the blue arithmetic progression to have distance $1$ (although they do not write this restriction in their papers).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 October 2025. View history", "references": "#188: [ErGr79][ErGr80]" }, { "number": 190, "url": "https://www.erdosproblems.com/190", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [], "formalized": "no", "statement": "Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\\{1,\\ldots,N\\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that\\[H(k)^{1/k}/k \\to \\infty\\]as $k\\to\\infty$?", "commentary": "This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemerédi's theorem, and it is easy to show that $H(k)^{1/k}\\to\\infty$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 October 2025. View history", "references": "#190: [ErGr79,p.333][ErGr80,p.17]" }, { "number": 193, "url": "https://www.erdosproblems.com/193", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [ "A231255" ], "formalized": "yes", "statement": "Let $S\\subseteq \\mathbb{Z}^3$ be a finite set and let $A=\\{a_1,a_2,\\ldots,\\}\\subset \\mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\\in S$ for all $i$. Must $A$ contain three collinear points?", "commentary": "Originally conjectured by Gerver and Ramsey [GeRa79], who showed that the answer is yes for $\\mathbb{Z}^2$, and for $\\mathbb{Z}^3$ that the largest number of collinear points can be bounded.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#193: [ErGr79][ErGr80]" }, { "number": 195, "url": "https://www.erdosproblems.com/195", "status": "open", "prize": "no", "tags": [ "arithmetic progressions" ], "oeis": [], "formalized": "yes", "statement": "What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\\cdotsj>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?", "commentary": "Davis, Entringer, Graham, and Simmons [DEGS77] have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.\n\nSee also [194] and [195].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#196: [ErGr79][ErGr80]" }, { "number": 197, "url": "https://www.erdosproblems.com/197", "status": "open", "prize": "no", "tags": [ "arithmetic progressions" ], "oeis": [], "formalized": "yes", "statement": "Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?", "commentary": "If three sets are allowed then this is possible.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#197: [ErGr79][ErGr80]" }, { "number": 200, "url": "https://www.erdosproblems.com/200", "status": "open", "prize": "no", "tags": [ "primes", "arithmetic progressions" ], "oeis": [ "A005115" ], "formalized": "yes", "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?", "commentary": "It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#200: [ErGr79][ErGr80]" }, { "number": 201, "url": "https://www.erdosproblems.com/201", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A003002", "A003003", "A003004", "A003005" ], "formalized": "no", "statement": "Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a $k$-term arithmetic progression? Is it true that\\[\\lim_{N\\to \\infty}\\frac{R_3(N)}{G_3(N)}=1?\\]", "commentary": "First asked and investigated by Riddell [Ri69]. It is trivial that $G_k(N)\\leq R_k(N)$, and it is possible that $G_k(N) 0$,\\[\\frac{N}{\\exp((\\log N)^{1/2+\\epsilon})} \\ll_\\epsilon f(N) < \\frac{N}{(\\log N)^c}\\]for some $c>0$. Erdős believed the lower bound is closer to the truth. \n\nThese bounds were improved by Croot [Cr03b] who proved\\[\\frac{N}{L(N)^{\\sqrt{2}+o(1)}}< f(N)<\\frac{N}{L(N)^{1/6-o(1)}},\\]where $L(N)=\\exp(\\sqrt{\\log N\\log\\log N})$. These bounds were further improved by Chen [Ch05] and then by de la Bretéche, Ford, and Vandehey [BFV13] to\\[\\frac{N}{L(N)^{1+o(1)}}0$ and large $n$,\\[s_{n+1}-s_n \\ll_\\epsilon s_n^{\\epsilon}?\\]Is it true that\\[s_{n+1}-s_n \\leq (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}?\\]", "commentary": "Erdős [Er51] showed that there are infinitely many $n$ such that\\[s_{n+1}-s_n > (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n},\\]so this bound would be the best possible. \n\nIn [Er79] Erdős says perhaps $s_{n+1}-s_n \\ll \\log s_n$, but he is 'very doubtful'.\n\nFilaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5+o(1)}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.\n\nGranville [Gr98] showed that $s_{n+1}-s_n\\ll_\\epsilon s_n^\\epsilon$ for all $\\epsilon>0$ follows from the ABC conjecture.\n\nSee also [489] and [145]. A more general form of this problem is given in [1101].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#208: [Er51][Er61][Er65b][Er79][Er81h,p.176]" }, { "number": 212, "url": "https://www.erdosproblems.com/212", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Is there a dense subset of $\\mathbb{R}^2$ such that all pairwise distances are rational?", "commentary": "Conjectured by Ulam. Erdős believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf [Sh18].\n\nIndeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw [SdZ10] then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.\n\nAscher, Braune, and Turchet [ABT20] observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).\n\nIn [Er87b] Erdős mentions that Besicovitch conjectured that the limit points of a rational distance set cannot contain arbitrarily large convex sets.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 October 2025. View history", "references": "#212: [Er61,p.246][Er75f,p.107][Er83c][Er87b,p.172]" }, { "number": 213, "url": "https://www.erdosproblems.com/213", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $n\\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?", "commentary": "Anning and Erdős [AnEr45] proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz [KK08], has $n=7$. \n\nAscher, Braune, and Turchet [ABT20] have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse [GIP24] have shown (unconditionally) that any such set must be very sparse, in that if $S\\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then\\[\\lvert S\\rvert \\ll (\\log N)^{O(1)}.\\]See also [130].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 October 2025. View history", "references": "#213: [Er75f,p.106][Er83c][Er87b,p.172]" }, { "number": 217, "url": "https://www.erdosproblems.com/217", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "For which $n$ are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?", "commentary": "An example with $n=4$ is an isosceles triangle with the point in the centre. Erdős originally believed this was impossible for $n\\geq 5$, but Pomerance constructed a set with $n=5$ (see [Er83c] for a description), and Palásti has proved such sets exist for all $n\\leq 8$.\n\nErdős believed this is impossible for all sufficiently large $n$. This would follow from $h(n)\\geq n$ for sufficiently large $n$, where $h(n)$ is as in [98].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 October 2025. View history", "references": "#217: [Er83c][Er87b,p.167][Er97e]" }, { "number": 218, "url": "https://www.erdosproblems.com/218", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A333230", "A333231", "A064113" ], "formalized": "yes", "statement": "Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.", "commentary": "In [Er85c] Erdős also conjectures that $d_n=d_{n+1}=\\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).\n\nBanks [Ba23] has given a heuristic argument, assuming a quantitative form of the prime tuples conjecture, that the first statement is true, and in fact for any $c\\geq 0$ and $\\epsilon>0$ the number of $n$ such that $p_n\\leq x$ and $d_{n+1}\\geq cd_n$ is\\[\\frac{\\pi(x)}{c+1}+O\\left(\\frac{x}{(\\log x)^{3/2+\\epsilon}}\\right).\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 January 2026. View history", "references": "#218: [Er55c][Er57][Er61][Er65b][Er85c]" }, { "number": 222, "url": "https://www.erdosproblems.com/222", "status": "open", "prize": "no", "tags": [ "number theory", "squares" ], "oeis": [ "A256435" ], "formalized": "no", "statement": "Let $n_10$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\\log x$ many consecutive primes $\\leq x$ such that the difference between any two is $>c_2$?", "commentary": "Erdős [Er49c] proved this is true for any $c_2>0$ if $c_1>0$ is sufficiently small (depending on $c_1$). \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 13 January 2026. View history", "references": "#238: [Er55c,p.7]" }, { "number": 241, "url": "https://www.erdosproblems.com/241", "status": "open", "prize": "$100", "tags": [ "additive combinatorics", "sidon sets" ], "oeis": [ "A387704" ], "formalized": "yes", "statement": "Let $f(N)$ be the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ such that the sums $a+b+c$ with $a,b,c\\in A$ are all distinct (aside from the trivial coincidences). Is it true that\\[ f(N)\\sim N^{1/3}?\\]", "commentary": "Originally asked to Erdős by Bose. Bose and Chowla [BoCh62] provided a construction proving one half of this, namely\\[(1+o(1))N^{1/3}\\leq f(N).\\]The best upper bound known to date is due to Green [Gr01],\\[f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3}\\](note that $(7/2)^{1/3}\\approx 1.519$). \n\nMore generally, Bose and Chowla conjectured that the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then\\[\\lvert A\\rvert \\sim N^{1/r}.\\]This is known only for $r=2$ (see [30]).\n\nThis is discussed in problem C11 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#241: [Er61][Er69][Er70b][Er70c][Er73][Er77c][Er80,p.99][ErGr80]" }, { "number": 242, "url": "https://www.erdosproblems.com/242", "status": "falsifiable", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A073101", "A075245", "A075246", "A075247", "A075248", "A287116" ], "formalized": "yes", "statement": "For every $n>2$ there exist distinct integers $1\\leq x0$.\nThis conjecture is equivalent (see Theorem 1 of [BlEl22]) to the statement that, for any prime $p$, there exist integers $a,c,d\\geq 1$ such that either $p\\equiv -a/c\\pmod{4acd-1}$ or $p\\equiv -\\frac{4c^2d+1}{k}\\pmod{4cd}$ for some $k\\mid 4c^2d+1$.\n Bright and Loughran [BrLo20] have shown there is no Brauer-Manin obstruction to the existence of solutions.\n If $f(n)$ counts the number of solutions then Elsholtz and Tao [ElTa13] have proved\\[\\sum_{p\\leq N}f(p)=N(\\log N)^{2+o(1)}\\]and $f(p)\\leq p^{3/5+o(1)}$ for all primes $p$.\n Elsholtz and Planitzer [ElPl20] have proved that for almost all $n$\\[f(n) \\geq (\\log n)^{\\log 6+o(1)}.\\]\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 January 2026. View history", "references": "#242: [Er50c][Er61][Er79][ErGr80][Va99,1.13]" }, { "number": 243, "url": "https://www.erdosproblems.com/243", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A000058" ], "formalized": "yes", "statement": "Let $1\\leq a_10.\\]A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1$ is known as Sylvester's sequence.\n\nDuverney [Du01] proved a weaker version of this problem: if\\[\\sum_{n\\geq 0}\\left(\\frac{a_{n+1}}{a_n^2}-1\\right)\\]converges then $\\sum \\frac{1}{a_n}$ is rational if and only if\\[a_{n}=a_{n-1}^2-a_{n-1}+1\\]for all large $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 21 January 2026. View history", "references": "#243: [ErGr80,p.64][Er88c,p.105]" }, { "number": 244, "url": "https://www.erdosproblems.com/244", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "yes", "statement": "Let $C>1$. Does the set of integers of the form $p+\\lfloor C^k\\rfloor$, for some prime $p$ and $k\\geq 0$, have density $>0$?", "commentary": "Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.\n\nDing [Di25] has proved that this is true for almost all $C>1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 October 2025. View history", "references": "#244: [Er61,p.230]" }, { "number": 247, "url": "https://www.erdosproblems.com/247", "status": "open", "prize": "no", "tags": [ "number theory", "irrationality" ], "oeis": [], "formalized": "yes", "statement": "Let $1\\leq a_1cn^2$ then $\\sum_{n=1}^\\infty \\frac{1}{2^{a_n}}$ is not the root of any quadratic polynomial'.\n\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 January 2026. View history", "references": "#247: [ErGr80,p.61][Er88c,p.106]" }, { "number": 249, "url": "https://www.erdosproblems.com/249", "status": "open", "prize": "no", "tags": [ "number theory", "irrationality" ], "oeis": [ "A256936" ], "formalized": "yes", "statement": "Is\\[\\sum_n \\frac{\\phi(n)}{2^n}\\]irrational? Here $\\phi$ is the Euler totient function.", "commentary": "The decimal expansion of this sum is A256936 on the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#249: [ErGr80,p.61][Er88c,p.102]" }, { "number": 251, "url": "https://www.erdosproblems.com/251", "status": "open", "prize": "no", "tags": [ "number theory", "irrationality" ], "oeis": [ "A098990" ], "formalized": "yes", "statement": "Is\\[\\sum \\frac{p_n}{2^n}\\]irrational? (Here $p_n$ is the $n$th prime.)", "commentary": "Erdős [Er58b] proved that $\\sum \\frac{p_n^k}{n!}$ is irrational for every $k\\geq 1$.\n\nIn [Er88c] he further conjectures that $\\sum \\frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\\geq 2$ and $g_n=o(p_n)$ then\\[\\sum_{n=1}^\\infty \\frac{p_n}{g_1\\cdots g_n}\\]is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)\n\nThe decimal expansion of this sum is A098990 on the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#251: [Er58b][ErGr80,p.62][Er88c,p.103]" }, { "number": 252, "url": "https://www.erdosproblems.com/252", "status": "open", "prize": "no", "tags": [ "number theory", "irrationality" ], "oeis": [ "A227988", "A227989" ], "formalized": "yes", "statement": "Let $k\\geq 1$ and $\\sigma_k(n)=\\sum_{d\\mid n}d^k$. Is\\[\\sum \\frac{\\sigma_k(n)}{n!}\\]irrational?", "commentary": "This is known now for $1\\leq k\\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erdős [Er52]. The case $k=3$ was proved independently by Schlage-Puchta [ScPu06] and Friedlander, Luca, and Stoiciu [FLC07]. The case $k=4$ was proved by Pratt [Pr22]. \n\nIt is known that this sum is irrational for all $k\\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta [ScPu06]) or Dickson's conjecture (Friedlander, Luca, and Stoiciu [FLC07]).\n\nThis is discussed in problem B14 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#252: [ErGr80,p.62][Er88c,p.102]" }, { "number": 254, "url": "https://www.erdosproblems.com/254", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that\\[\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty\\]and\\[\\sum_{n\\in A} \\{ \\theta n\\}=\\infty\\]for every $\\theta\\in (0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.", "commentary": "Cassels [Ca60] proved this under the alternative hypotheses\\[\\lim \\frac{\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty\\]and\\[\\sum_{n\\in A} \\{ \\theta n\\}^2=\\infty\\]for every $\\theta\\in (0,1)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 December 2025. View history", "references": "#254: [Er61]" }, { "number": 256, "url": "https://www.erdosproblems.com/256", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Let $n\\geq 1$ and $f(n)$ be maximal such that for any integers $1\\leq a_1\\leq \\cdots \\leq a_n$ we have\\[\\max_{\\lvert z\\rvert=1}\\left\\lvert \\prod_{i}(1-z^{a_i})\\right\\rvert\\geq f(n).\\]Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that\\[\\log f(n) \\gg n^c?\\]", "commentary": "Erdős and Szekeres [ErSz59] proved that $\\lim f(n)^{1/n}=1$ and $f(n)>\\sqrt{2n}$. Erdős proved an upper bound of $\\log f(n) \\ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Atkinson [At61] showed that $\\log f(n) \\ll n^{1/2}\\log n$.\n\nThis was improved to\\[\\log f(n) \\ll n^{1/3}(\\log n)^{4/3}\\]by Odlyzko [Od82].\n\nIf we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\\cdots0\\]for some $\\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.\n\nThis problem was originally wrongly stated without the assumption that the sequence be increasing; DeepMind has given a counterexample without this assumption (see the comments).\n\nKoizumi [Ko25c] has proved that $a_n=\\lfloor \\alpha^{2^n}\\rfloor$ is an irrationality sequence of this type for all but countably many $\\alpha>1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 April 2026. View history", "references": "#263: [ErGr80,p.63][Er88c,p.105]" }, { "number": 264, "url": "https://www.erdosproblems.com/264", "status": "open", "prize": "no", "tags": [ "irrationality" ], "oeis": [], "formalized": "yes", "statement": "Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\\neq 0$ and $b_n\\neq 0$ for all $n$) the sum\\[\\sum \\frac{1}{a_n+b_n}\\]is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?", "commentary": "A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In [ErGr80] they also ask whether such a sequence can have polynomial growth, but Erdős later retracted this in [Er88c], claiming 'It is not hard to show that it cannot increase slower than exponentially'.\n\nKovač and Tao [KoTa24] have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\\sum\\frac{1}{a_n}$ converges and\\[\\liminf \\left(a_n^2\\sum_{k>n}\\frac{1}{a_k^2}\\right) >0 \\]is not such an irrationality sequence. In particular, any strictly increasing sequence with $\\limsup a_{n+1}/a_n <\\infty$ is not such an irrationality sequence.\n\nOn the other hand, Kovač and Tao do prove that for any function $F$ with $\\lim F(n+1)/F(n)=\\infty$ there exists such an irrationality sequence with $a_n\\sim F(n)$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 January 2026. View history", "references": "#264: [ErGr80,p.63][Er88c,p.105]" }, { "number": 265, "url": "https://www.erdosproblems.com/265", "status": "open", "prize": "no", "tags": [ "irrationality" ], "oeis": [], "formalized": "no", "statement": "Let $1\\leq a_11$.\n\nIt remains open whether one can achieve\\[\\limsup a_n^{1/2^n}>1.\\]A folklore result states that $\\sum \\frac{1}{a_n}$ is irrational whenever $\\lim a_n^{1/2^n}=\\infty$, and hence such a sequence cannot grow faster than doubly exponentially - the remaining question is the precise exponent possible.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 21 January 2026. View history", "references": "#265: [ErGr80,p.64][Er88c,p.104]" }, { "number": 267, "url": "https://www.erdosproblems.com/267", "status": "open", "prize": "no", "tags": [ "irrationality" ], "oeis": [], "formalized": "yes", "statement": "Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_11$. Must\\[\\sum_k\\frac{1}{F_{n_k}}\\]be irrational?", "commentary": "It may be sufficient to have $n_k/k\\to \\infty$. Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\\sum \\frac{1}{F_{2^n}}$ is irrational - in fact,\\[\\sum \\frac{1}{F_{2^n}}=\\frac{7-\\sqrt{5}}{2}.\\]Badea [Ba87] proved that $\\sum \\frac{1}{F_{2^n+1}}$ is irrational. \n\nThe sum $\\sum \\frac{1}{F_n}$ itself was proved to be irrational by André-Jeannin [An89].\n\nThe main problem has been proved for $c\\geq 2$ by Badea [Ba93]. It remains open for $10$, $a_k\\leq (\\frac{1}{2}+\\epsilon)k^2$ for all sufficiently large $k$. van Doorn and Sothanaphan have noted in the comment section that Moy's proof can be upgraded to give a fully explicit result of\\[a_k\\leq \\frac{(k-1)(k+2)}{2}+n\\]for all $k\\geq 0$.\n\nIn general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 January 2026. View history", "references": "#271: [ErGr80,p.22]" }, { "number": 272, "url": "https://www.erdosproblems.com/272", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [], "formalized": "no", "statement": "Let $N\\geq 1$. What is the largest $t$ such that there are $A_1,\\ldots,A_t\\subseteq \\{1,\\ldots,N\\}$ with $A_i\\cap A_j$ a non-empty arithmetic progression for all $i\\neq j$?", "commentary": "Simonovits and Sós [SiSo81] have shown that $t\\ll N^2$.\n\nErdős and Graham asked whether the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\\{1,\\ldots,N\\}$ containing some fixed element, 'presumably the integer $\\lfloor N/2\\rfloor$'. This was disproved by Simonovits and Sós [SiSo81], who observed that taking all sets containing at most $3$ elements, containing some fixed element, produces $\\binom{N}{2}+1$ many such sets, which is asymptotically greater than the number of arithmetic progressions containing a fixed element, which is $\\sim \\frac{\\pi^2}{24}N^2$.\n\nIf we drop the non-empty requirement then Graham, Simonovits, and Sós [GSS80] have shown that\\[t\\leq \\binom{N}{3}+\\binom{N}{2}+\\binom{N}{1}+1\\]and this is best possible.\n\nSzabo [Sz99] proved that the maximal such $t$ is equal to\\[\\frac{N^2}{2}+O(N^{5/3}(\\log N)^3),\\]resolving the asymptotic question. On the other hand, Szabo showed that the conjecture of Simonovits and Sós that $\\binom{n}{2}+1$ is best possible is false, giving a construction which yields\\[t \\geq \\binom{N}{2}+\\left\\lfloor\\frac{N-1}{4}\\right\\rfloor+1.\\]Szabo conjectures that the asymptotic $t=\\binom{N}{2}+O(N)$ holds, and that in any extremal example there is an integer contained in all sets.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#272: [ErGr80,p.20]" }, { "number": 273, "url": "https://www.erdosproblems.com/273", "status": "open", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "yes", "statement": "Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\\geq 5$?", "commentary": "Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#273: [ErGr80,p.24]" }, { "number": 274, "url": "https://www.erdosproblems.com/274", "status": "open", "prize": "no", "tags": [ "group theory", "covering systems" ], "oeis": [], "formalized": "yes", "statement": "If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)", "commentary": "A question of Herzog and Schönheim, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.\n\nThis conjecture was proved in the case when all the $G_i$ are subnormal in $G$ by Sun [Su04]. In particular if $G$ is abelian (which was the special case asked about in [Er77c] and [ErGr80]) the answer to the original question is no.\n\nMargolis and Schnabel [MaSc19] proved this conjecture for all groups $G$ of size $<1440$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#274: [Er77c,p.49][ErGr80,p.26][Er97c,p.53]" }, { "number": 276, "url": "https://www.erdosproblems.com/276", "status": "open", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "yes", "statement": "Is there an infinite Lucas sequence $a_0,a_1,\\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?", "commentary": "Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham [Gr64] showed that\\[a_0 = 1786772701928802632268715130455793\\]and\\[a_1 = 1059683225053915111058165141686995\\]generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'. \n\nThis problem has been 'conjecturally solved' by Ismailescu and Son [IsSo14], in that they provide an explicit infinite Lucas sequence in which all the terms are composite, and believe that no covering system is responsible for this. See the comment by van Doorn below for more details.\n\nSee also [1113] for another problem in which the question is whether covering systems are always responsible.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 December 2025. View history", "references": "#276: [ErGr80,p.27]" }, { "number": 278, "url": "https://www.erdosproblems.com/278", "status": "open", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "no", "statement": "Let $A=\\{n_1<\\cdots0$ (provided $m$ is taken sufficiently large depending on $\\alpha$).\n\nCassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\\geq 1$ and if we allow $n_i=n_j$. \n\nAlekseyev [Al19] has proved this when $p(x)=x^2$, for all $m>8542$. For example,\\[1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12}\\]and\\[200 = 2^2+4^2+6^2+12^2.\\]van Doorn [vD25] has investigated the question of what 'sufficiently large' means for $p(x)=x$. van Doorn has also proved the original conjecture for many linear and quadratic polynomials, for example $p(x)=x+5$ or $p(x)=x^2+100$ - see the comments section.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#283: [ErGr80,p.32]" }, { "number": 287, "url": "https://www.erdosproblems.com/287", "status": "falsifiable", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 2$. Is it true that, for any distinct integers $11$ both occur for infinitely many $n$?", "commentary": "Steinerberger has observed that the answer to the second question is trivially yes: for example, any $n$ which begins with a $2$ in base $3$ has $3\\mid (a_n,L_n)$.\n\nMore generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\\cdots+\\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. This can be seen by writing\\[a_n = \\frac{L_n}{1}+\\cdots+\\frac{L_n}{n}\\]and observing that the right-hand side is congruent to $1+\\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows immediately from Wolstenholme's theorem.)\n\nThis leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of $\\asymp\\frac{x}{\\log x}$ for the number of $n\\in [1,x]$ such that $(a_n,L_n)=1$. In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.\n\nWu and Yan [WuYa22] have proved, conditional on $\\frac{1}{\\log p}$ being linearly independent over $\\mathbb{Q}$ for any finite collection of primes $p$ (itself a consequence of Schanuel's conjecture), that the set of $n$ for which $(a_n,L_n)>1$ has upper density $1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 12 January 2026. View history", "references": "#291: [ErGr80,p.34]" }, { "number": 293, "url": "https://www.erdosproblems.com/293", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to\\[1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}\\]with $1\\leq n_1<\\cdots 0$, and noted a close connection to [304]. In particular, if $N(b)\\ll \\log\\log b$ as in [304] then it is likely the methods of [vDTa25b] prove $v(k) \\geq e^{e^{ck}}$ for some $c>0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 December 2025. View history", "references": "#293: [ErGr80,p.35]" }, { "number": 295, "url": "https://www.erdosproblems.com/295", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [], "formalized": "yes", "statement": "Let $N\\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\\leq n_1<\\cdots 0$ such that\\[-c < k(N)-(e-1)N \\ll \\frac{N}{\\log N}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#295: [ErGr80]" }, { "number": 301, "url": "https://www.erdosproblems.com/301", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A390394" ], "formalized": "no", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to\\[\\frac{1}{a}= \\frac{1}{b_1}+\\cdots+\\frac{1}{b_k}\\]with distinct $a,b_1,\\ldots,b_k\\in A$?Estimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?", "commentary": "The example $A=(N/2,N]\\cap \\mathbb{N}$ shows that $f(N)\\geq N/2$.\n\nWouter van Doorn has given an elementary argument that proves\\[f(N)\\leq (25/28+o(1))N.\\]Indeed, consider the sets $S_a=\\{2a,3a,4a,6a,12a\\}\\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\\leq N/12$ and at least one element of $S_a$ when $N/120}$ with $b$ squarefree. Are there integers $10$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 January 2026. View history", "references": "#311: [ErGr80,p.40]" }, { "number": 312, "url": "https://www.erdosproblems.com/312", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [], "formalized": "yes", "statement": "Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\\sum_{n\\in A}\\frac{1}{n}>K$ there exists some $S\\subseteq A$ such that\\[1-e^{-cK} < \\sum_{n\\in S}\\frac{1}{n}\\leq 1?\\]", "commentary": "Erdős and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 January 2026. View history", "references": "#312: [ErGr80]" }, { "number": 313, "url": "https://www.erdosproblems.com/313", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A054377" ], "formalized": "yes", "statement": "Are there infinitely many solutions to\\[\\frac{1}{p_1}+\\cdots+\\frac{1}{p_k}=1-\\frac{1}{m},\\]where $m\\geq 2$ is an integer and $p_1<\\cdots0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with\\[0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}?\\]Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$,\\[\\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]}\\]whenever the left-hand side is not zero?", "commentary": "Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example\\[\\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}.\\]Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of\\[2^{-n\\frac{(\\log\\log\\log n)^{1+o(1)}}{\\log n}},\\]and van Doorn gives a heuristic that suggests this may be the true order of magnitude.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 January 2026. View history", "references": "#317: [ErGr80,p.42]" }, { "number": 319, "url": "https://www.erdosproblems.com/319", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [], "formalized": "yes", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that\\[\\sum_{n\\in A}\\frac{\\delta_n}{n}=0\\]and\\[\\sum_{n\\in A'}\\frac{\\delta_n}{n}\\neq 0\\]for all non-empty $A'\\subsetneq A$?", "commentary": "Adenwalla has observed that a lower bound of\\[\\lvert A\\rvert\\geq (1-\\tfrac{1}{e}+o(1))N\\]follows from the main result of Croot [Cr01], which states that there exists some set of integers $B\\subset [(\\frac{1}{e}-o(1))N,N]$ such that $\\sum_{b\\in B}\\frac{1}{b}=1$. Since the sum of $\\frac{1}{m}$ for $m\\in [c_1N,c_2N]$ is asymptotic to $\\log(c_2/c_1)$ we must have $\\lvert B\\rvert \\geq (1-\\tfrac{1}{e}+o(1))N$.\n\nWe may then let $A=B\\cup\\{1\\}$ and choose $\\delta(n)=-1$ for all $n\\in B$ and $\\delta(1)=1$. \n\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#319: [ErGr80]" }, { "number": 320, "url": "https://www.erdosproblems.com/320", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A072207" ], "formalized": "no", "statement": "Let $S(N)$ count the number of distinct sums of the form $\\sum_{n\\in A}\\frac{1}{n}$ for $A\\subseteq \\{1,\\ldots,N\\}$. Estimate $S(N)$.", "commentary": "Bleicher and Erdős [BlEr75] proved the lower bound\\[\\log S(N)\\geq \\frac{N}{\\log N}\\left(\\log 2\\prod_{i=3}^k\\log_iN\\right),\\]valid for $k\\geq 4$ and $\\log_kN\\geq k$, and also [BlEr76b] proved the upper bound\\[\\log S(N)\\leq \\frac{N}{\\log N}\\left(\\log_r N \\prod_{i=3}^r \\log_iN\\right),\\]valid for $r\\geq 1$ and $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\n\nBettin, Grenié, Molteni, and Sanna [BGMS25] improved the lower bound to\\[\\log S(N) \\geq \\frac{N}{\\log N}\\left(2\\log 2\\left(1-\\frac{3/2}{\\log_kN}\\right)\\prod_{i=3}^k\\log_iN\\right),\\]valid for $k\\geq 4$ and $\\log_kN\\geq 3/2$. (In particular this goes to infinity faster than the lower bound of Bleicher and Erdős.)\n\nSee also [321].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#320: [ErGr80,p.43]" }, { "number": 321, "url": "https://www.erdosproblems.com/321", "status": "open", "prize": "no", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A384927", "A391592" ], "formalized": "yes", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that all sums $\\sum_{n\\in S}\\frac{1}{n}$ are distinct for $S\\subseteq A$?", "commentary": "Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that\\[\\frac{N}{\\log N}\\prod_{i=3}^k\\log_iN\\leq R(N)\\leq \\frac{1}{\\log 2}\\log_r N\\left(\\frac{N}{\\log N} \\prod_{i=3}^r \\log_iN\\right),\\]valid for any $k\\geq 4$ with $\\log_kN\\geq k$ and any $r\\geq 1$ with $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\n\nSee also [320].\n\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#321: [ErGr80,p.43]" }, { "number": 322, "url": "https://www.erdosproblems.com/322", "status": "open", "prize": "no", "tags": [ "number theory", "powers" ], "oeis": [ "A025456", "A025418" ], "formalized": "no", "statement": "Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that\\[1_A^{(k)}(n) >n^c?\\]", "commentary": "Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\\leq n^{o(1)}$, but this was disproved by Mahler [Ma36] for $k=3$, who constructed infinitely many $n$ such that\\[1_A^{(3)}(n)\\gg n^{1/12}\\](where $A$ is the set of cubes). Erdős believed Hypothesis $K$ fails for all $k\\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\\epsilon>0$\\[\\sum_{n\\leq N}1_A^{(k)}(n)^2 \\ll_\\epsilon N^{1+\\epsilon}.\\]Erdős and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.'\n\nIndependently Erdős [Er36] and Chowla proved that for all $k\\geq 3$ and infinitely many $n$\\[1_A^{(k)}(n) \\gg n^{c/\\log\\log n}\\]for some constant $c>0$ (depending on $k$). In [Er65b] Erdős claims an unpublished proof that, if $B$ is the set of $k$th powers of any set of positive density, then\\[\\limsup 1_B^{(k)}(n)=\\infty.\\]This is discussed in problem D4 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#322: [Er65b][ErGr80]" }, { "number": 323, "url": "https://www.erdosproblems.com/323", "status": "open", "prize": "no", "tags": [ "number theory", "powers" ], "oeis": [], "formalized": "yes", "statement": "Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that\\[f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon}\\]for all $\\epsilon>0$? Is it true that if $m0$.\n\nFor $k>2$ it is not known if $f_{k,k}(x)=o(x)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#323: [ErGr80]" }, { "number": 324, "url": "https://www.erdosproblems.com/324", "status": "open", "prize": "no", "tags": [ "number theory", "powers" ], "oeis": [], "formalized": "yes", "statement": "Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $af(n)$.)", "commentary": "Erdős originally asked if even $f(n)\\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog [Ba89] who proved that\\[f(n) \\ll_\\epsilon n^{\\frac{4}{9\\sqrt{e}}+\\epsilon}\\]for all $\\epsilon>0$. (Note $\\frac{4}{9\\sqrt{e}}=0.2695\\ldots$.)\n\nIt is likely that $f(n)\\leq n^{o(1)}$.\n\nSee also Problem 59 on Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 April 2026. View history", "references": "#334: [Er76e,p.272][ErGr80,p.70][Er82d,p.55]" }, { "number": 335, "url": "https://www.erdosproblems.com/335", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $d(A)$ denote the density of $A\\subseteq \\mathbb{N}$. Characterise those $A,B\\subseteq \\mathbb{N}$ with positive density such that\\[d(A+B)=d(A)+d(B).\\]", "commentary": "One way this can happen is if there exists $\\theta>0$ such that\\[A=\\{ n>0 : \\{ n\\theta\\} \\in X_A\\}\\textrm{ and }B=\\{ n>0 : \\{n\\theta\\} \\in X_B\\}\\]where $\\{x\\}$ denotes the fractional part of $x$ and $X_A,X_B\\subseteq \\mathbb{R}/\\mathbb{Z}$ are such that $\\mu(X_A+X_B)=\\mu(X_A)+\\mu(X_B)$. Are all possible $A$ and $B$ generated in a similar way (using other groups)?\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#335: [ErGr80,p.51]" }, { "number": 336, "url": "https://www.erdosproblems.com/336", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "no", "statement": "For $r\\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\\subseteq \\mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). Find the value of\\[\\lim_r \\frac{h(r)}{r^2}.\\]", "commentary": "A simple example of the order of a basis differing from the exact order is given by $A=\\cup_{k\\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$.\n\nErdős and Graham [ErGr80b] have shown that a basis $A$ has an exact order if and only if $a_2-a_1,a_3-a_2,a_4-a_3,\\ldots$ are coprime. They also proved that\\[\\frac{1}{4}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{5}{4}.\\]The best bounds known for the limit are\\[\\frac{1}{3}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{1}{2},\\]the lower bound originally due to Grekos [Gr88] and the upper bound to Nash [Na93]. Improved bounds in the lower order terms were given by Plagne [Pl04].\n\nErdős and Graham [ErGr80b] showed $h(2)=4$. Nash [Na93] showed $h(3)=7$. The value of $h(4)$ is unknown, but it is known [Pl04] that $10\\leq h(4)\\leq 11$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 October 2025. View history", "references": "#336: [ErGr80,p.51]" }, { "number": 338, "url": "https://www.erdosproblems.com/338", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "no", "statement": "The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?", "commentary": "Bateman has observed that for $h\\geq 3$ there is a basis of order $h$ with no restricted order, taking\\[A=\\{1\\}\\cup \\{x>0 : h\\mid x\\}.\\]Kelly [Ke57] has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additional assumption that the basis has positive lower density). Kelly's conjecture was disproved by Hennecart [He05], who constructed a basis of order $2$ with restricted order $4$.\n\nThe set of squares has order $4$ and restricted order $5$ (see [Pa33]) and the set of triangular numbers has order $3$ and restricted order $3$ (see [Sc54]).\n\nIs it true that if $A\\backslash F$ is a basis for all finite sets $F$ then $A$ must have a restricted order? What if they are all bases of the same order?\n\nHegyvári, Hennecart, and Plagne [HHP07] have shown that for all $k\\geq2$ there exists a basis of order $k$ which has restricted order at least\\[2^{k-2}+k-1.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 September 2025. View history", "references": "#338: [ErGr80][ErGr80b]" }, { "number": 340, "url": "https://www.erdosproblems.com/340", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics", "sidon sets" ], "oeis": [ "A080200", "A005282" ], "formalized": "yes", "statement": "Let $A=\\{1,2,4,8,13,21,31,45,66,81,97,\\ldots\\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that\\[\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2-\\epsilon}\\]for all $\\epsilon>0$ and large $N$?", "commentary": "This sequence is sometimes called the Mian-Chowla sequence. It is trivial that this sequence grows at least like $\\gg N^{1/3}$. \n\nErdős and Graham [ErGr80] also asked about the difference set $A-A$, whether this has positive density, and whether this contains $22$. It does contain $22$, since $a_{15}-a_{14}=204-182=22$. The smallest integer which is unknown to be in $A-A$ is $33$ (see A080200). It may be true that all or almost all integers are in $A-A$.\n\nThis sequence is at OEIS A005282.\n\nSee also [156].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 November 2025. View history", "references": "#340: [ErGr80,p.53]" }, { "number": 341, "url": "https://www.erdosproblems.com/341", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A=\\{a_1<\\cdotsa_n$ which can be expressed uniquely as $a_i+a_j$ for $iT(n^{k+1})$?", "commentary": "Erdős and Graham [ErGr80] remark that very little is known about $T(A)$ in general. It is known that\\[T(n)=1, T(n^2)=128, T(n^3)=12758,\\]\\[T(n^4)=5134240,\\textrm{ and }T(n^5)=67898771.\\]Erdős and Graham remark that a good candidate for the $n$ in the question are $k=2^t$ for large $t$, perhaps even $t=3$, because of the highly restricted values of $n^{2^t}$ modulo $2^{t+1}$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#345: [ErGr80,p.55]" }, { "number": 346, "url": "https://www.erdosproblems.com/346", "status": "open", "prize": "no", "tags": [ "number theory", "complete sequences" ], "oeis": [], "formalized": "yes", "statement": "Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that\n $A\\backslash B$ is complete for any finite subset $B$ and \n $A\\backslash B$ is not complete for any infinite subset $B$.\n(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)Is it true that if $a_{n+1}/a_n \\geq 1+\\epsilon$ for some $\\epsilon>0$ and all $n$ then\\[\\lim_n \\frac{a_{n+1}}{a_n}=\\frac{1+\\sqrt{5}}{2}?\\]", "commentary": "Graham [Gr64d] has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erdős and Graham [ErGr80] remark that it is easy to see that if $a_{n+1}/a_n>\\frac{1+\\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 December 2025. View history", "references": "#346: [ErGr80,p.57]" }, { "number": 348, "url": "https://www.erdosproblems.com/348", "status": "open", "prize": "no", "tags": [ "number theory", "complete sequences" ], "oeis": [], "formalized": "yes", "statement": "For what values of $0\\leq m0$ and all $1<\\alpha < \\frac{1+\\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even know whether $\\lfloor (3/2)^n\\rfloor$ is odd or even infinitely often.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#349: [ErGr80,p.57]" }, { "number": 351, "url": "https://www.erdosproblems.com/351", "status": "open", "prize": "no", "tags": [ "number theory", "complete sequences" ], "oeis": [], "formalized": "yes", "statement": "Let $p(x)\\in \\mathbb{Q}[x]$. Is it true that\\[A=\\{ p(n)+1/n : n\\in \\mathbb{N}\\}\\]is strongly complete, in the sense that, for any finite set $B$,\\[\\left\\{\\sum_{n\\in X}n : X\\subseteq A\\backslash B\\textrm{ finite }\\right\\}\\]contains all sufficiently large integers?", "commentary": "Graham [Gr63] proved this is true when $p(n)=n$. Erdős and Graham also ask which rational functions $r(x)\\in\\mathbb{Z}(x)$ force $\\{ r(n) : n\\in\\mathbb{N}\\}$ to be complete?\n\nGraham [Gr64f] gave a complete characterisation of which polynomials $r\\in \\mathbb{R}[x]$ are such that $\\{ r(n) : n\\in \\mathbb{N}\\}$ is complete.\n\nIn the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from [Gr63] together with result of Alekseyev [Al19] mentioned in [283].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#351: [ErGr80,p.58]" }, { "number": 352, "url": "https://www.erdosproblems.com/352", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "yes", "statement": "Is there some $c>0$ such that every measurable $A\\subseteq \\mathbb{R}^2$ of measure $\\geq c$ contains the vertices of a triangle of area 1?", "commentary": "Erdős (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in [Er78d] and [Er83d] it 'follows easily from the Lebesgue density theorem'). \n\nIn [Er78d] and [Er83d] he speculated that perhaps $C=4\\pi/\\sqrt{27}\\approx 2.418$ works, which would be the best possible, as witnessed by a circle of radius $<2\\cdot 3^{-3/4}$.\n\nFurther evidence for this is given by a result of Freiling and Mauldin [Ma02], who proved that if $A$ has outer measure $>4\\pi/\\sqrt{27}$ then $A$ contains the vertices of a triangle with area $>1$. This also proves the same threshold for the original problem under the assumption that $A$ is a compact convex set. \n\nMauldin also discusses this problem in [Ma13], in which he notes that it suffices to prove this under the assumption that $A$ is the union of the interiors of $n<\\infty$ many compact convex sets. Freiling and Mauldin (see [Ma13]) have proved this conjecture if $1\\leq n\\leq 3$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#352: [Er78d,p.122][Er81b,p.30][Er83d,p.323][Va99,2.47]" }, { "number": 354, "url": "https://www.erdosproblems.com/354", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset\\[\\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\}\\]complete? That is, can all sufficiently large natural numbers $n$ be written as\\[n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor\\]for some finite $S,T\\subset \\mathbb{N}$?What if $2$ is replaced by some $\\gamma\\in(1,2)$?", "commentary": "This question was first mentioned by Graham [Gr71]. \n\nHegyvári [He89] proved that this holds if $\\alpha=m/2^n$ is a dyadic rational and $\\beta$ is not. He later [He91] proved that, for any fixed $\\alpha>0$, the set of $\\beta$ for which this holds either has measure $0$ or infinite measure. In [He94] he proved that the set of $(\\alpha,\\beta)$ for which the corresponding set of sums does not contain an infinite arithmetic progression has cardinality continuum.\n\nHegyvári [He89] proved that the sequence is not complete if $\\alpha\\geq 2$ and $\\beta =2^k\\alpha$ for some $k\\geq 0$. Jiang and Ma [JiMa24] and Fang and He [FaHe25] prove that the sequence is not complete if $1<\\alpha<2$ and $\\beta=2^k\\alpha$ for some sufficiently large $k$.\n\nIt is likely (and Hegyvári conjectures) that the assumption $\\alpha/\\beta$ irrational can be weakened to $\\alpha/\\beta \\neq 2^k$ and either $\\alpha$ or $\\beta$ not a dyadic rational.\n\nIn the comments van Doorn proves the sequence is complete if $\\alpha < 2<\\beta<3$, and also proves that if either $\\alpha$ or $\\beta$ is not a dyadic rational then the corresponding sequence with ceiling functions replacing the floor functions is complete.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 December 2025. View history", "references": "#354: [ErGr80,p.58]" }, { "number": 357, "url": "https://www.erdosproblems.com/357", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A364132", "A364153" ], "formalized": "yes", "statement": "Let $1\\leq a_1<\\cdots 0$?", "commentary": "A problem of MacMahon, studied by Andrews [An75]. When $n=1$ this sequence begins\\[1,2,4,5,8,10,14,15,\\ldots.\\]This sequence is A002048 in the OEIS. Andrews conjectures\\[a_k\\sim \\frac{k\\log k}{\\log\\log k}.\\]Porubsky [Po77] proved that, for any $\\epsilon>0$, there are infinitely many $k$ such that\\[a_k < (\\log k)^\\epsilon \\frac{k\\log k}{\\log\\log k},\\]and also that if $A(x)$ counts the number of $a_i\\leq x$ then\\[\\limsup \\frac{A(x)}{\\pi(x)}\\geq \\frac{1}{\\log 2}\\]where $\\pi(x)$ counts the number of primes $\\leq x$.\n\nSee also [839].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#359: [Er77c,p.70][Er78f,p.9][ErGr80,p.59 and p.94]" }, { "number": 361, "url": "https://www.erdosproblems.com/361", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\\subseteq \\{1,\\ldots,\\lfloor cn\\rfloor\\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?", "commentary": "View the LaTeX source\n\n\n\n\n \n This page was last edited 17 October 2025. View history", "references": "#361: [ErGr80,p.59]" }, { "number": 364, "url": "https://www.erdosproblems.com/364", "status": "verifiable", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A060355", "A076445" ], "formalized": "yes", "statement": "Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\\mid n$ then $p^2\\mid n$)?", "commentary": "Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=8y^2+1$. \n\nThis conjecture was also made by Mollin and Walsh [MoWa86]. Erdős [Er76d] believed the answer to this question is no, and in fact if $n_k$ is the $k$th powerful number then\\[n_{k+2}-n_k > n_k^c\\]for some constant $c>0$. The abc conjecture implies there are only finitely many such triples.\n\nIt is trivial that there are no quadruples of consecutive powerful numbers since one must be $2\\pmod{4}$. \n\nChan [Ch25] has shown there are no triples $n-1,n,n+1$ of powerful numbers with $n$ a cube. \n\nBy OEIS A060355 there are no such $n$ for $n<10^{22}$.\n\nSee also [137], [365], and [938].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 20 December 2025. View history", "references": "#364: [Er76d][ErGr80,p.68]" }, { "number": 365, "url": "https://www.erdosproblems.com/365", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A060355", "A060859", "A175155" ], "formalized": "no", "statement": "Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?Is the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?", "commentary": "Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$. \n\nThe list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.\n\nThe answer to the first question is no: Golomb [Go70] observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker [Wa76] proved that the equation\\[7^3x^2=3^3y^2+1\\]has infinitely many solutions, giving infinitely many counterexamples.\n\nSee also [364].\n\nThis is discussed in problem B16 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#365: [Er76d,p.31][ErGr80,p.68]" }, { "number": 366, "url": "https://www.erdosproblems.com/366", "status": "verifiable", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A060355" ], "formalized": "yes", "statement": "Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\\mid n$ then $p^2\\mid n$ and if $p\\mid n+1$ then $p^3\\mid n+1$.", "commentary": "Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=8y^2+1$. \n\nNote that $8$ is 3-full and $9$ is $2$-full. Erdős and Graham asked if this is the only pair of such consecutive integers. Stephan has observed that $12167=23^3$ and $12168=2^33^213^2$ (a pair already known to Golomb [Go70]) is another example, but (by OEIS A060355) there are no other examples for $n<10^{22}$.\n\nIn [Er76d] Erdős asks the weaker question of whether there are any consecutive pairs of $3$-full integers (which is also discussed in problem B16 of Guy's collection [Gu04]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#366: [Er76d,p.31][ErGr80,p.68]" }, { "number": 367, "url": "https://www.erdosproblems.com/367", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A057521" ], "formalized": "no", "statement": "Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$,\\[\\prod_{n\\leq m0$,\\[\\limsup \\frac{\\prod_{n\\leq m0$, there are infinitely many $n$ such that $F(n) <(\\log n)^{2+\\epsilon}$.\n\nPasten [Pa24b] has proved that\\[F(n) \\gg \\frac{(\\log\\log n)^2}{\\log\\log\\log n}.\\]The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#368: [Er65b,p.218][Er76d,p.27][ErGr80,p.69]" }, { "number": 371, "url": "https://www.erdosproblems.com/371", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A070089" ], "formalized": "yes", "statement": "Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n) (0.2017-o(1))x,\\]and the same lower bound for the complement.\n\nIn [Er79e] Erdős also asks whether, for every $\\alpha$, the density of the set of $n$ where\\[P(n+1)>P(n)n^\\alpha\\]exists.\n\nTeräväinen [Te18] has proved that the logarithmic density of the set of $n$ for which $P(n)P(n)n^\\alpha$ exists and is equal to\\[\\int_{[0,1]^2}1_{y\\geq x+\\alpha}u(x)u(y)\\mathrm{d}x\\mathrm{d}y\\]where $u(x)=x^{-1}\\rho(x^{-1}-1)$ and $\\rho$ is the Dickman function. Wang [Wa21] has proved the same value holds for the asymptotic density (and in particular provided an affirmative answer to the original question) conditional on the Elliott-Halberstam conjecture for friable integers.\n\nThe sequence of such $n$ is A070089 in the OEIS.\n\nSee also [372] and [928].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#371: [ErPo78,p.320][Er79e][ErGr80,p.70][Er85c,p.82][Va99,1.10]" }, { "number": 373, "url": "https://www.erdosproblems.com/373", "status": "open", "prize": "no", "tags": [ "number theory", "factorials" ], "oeis": [ "A003135" ], "formalized": "yes", "statement": "Show that the equation\\[n! = a_1!a_2!\\cdots a_k!,\\]with $n-1>a_1\\geq a_2\\geq \\cdots \\geq a_k\\geq 2$, has only finitely many solutions.", "commentary": "This would follow if $P(n(n+1))/\\log n\\to \\infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem [368]). Erdős [Er76d] proved that this problem would also follow from showing that $P(n(n-1))>4\\log n$.\n\nThe condition $a_1a_1\\geq a_2$ then\\[a_1\\geq n-5\\log\\log n,\\]and says it 'would be nice' to prove $a_1\\geq n-o(\\log\\log n)$. Bhat and Ramachandra [BhRa10] replace the $5$ with $(1+o(1))\\frac{1}{\\log 2}$, and also prove that the same bound holds for arbitrary $k\\geq 2$.\n\nNumerical investigations on solutions to $n!=a_1!a_2!$ have been carried out by Caldwell [Ca94] and Habsieger [Ha19], and it is known that there are no solutions aside from $10!=6!7!$ for $n\\leq 10^{3000}$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 January 2026. View history", "references": "#373: [Er76d,p.28][ErGr80,p.70][Er97e,p.537]" }, { "number": 374, "url": "https://www.erdosproblems.com/374", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A388851", "A387184", "A389117", "A389148" ], "formalized": "no", "statement": "For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1<\\cdots 1\\}$,\n $\\lvert D_3\\cap \\{1,\\ldots,n\\}\\rvert = o(\\lvert D_4\\cap \\{1,\\ldots,n\\}\\rvert)$,\n the least element of $D_6$ is $527$, and\n $D_k=\\emptyset$ for $k>6$.\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#374: [ErGr76][ErGr80]" }, { "number": 375, "url": "https://www.erdosproblems.com/375", "status": "falsifiable", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that for any $n,k\\geq 1$, if $n+1,\\ldots,n+k$ are all composite then there are distinct primes $p_1,\\ldots,p_k$ such that $p_i\\mid n+i$ for $1\\leq i\\leq k$?", "commentary": "Note this is trivial when $k\\leq 2$. Originally conjectured by Grimm [Gr69]. This is a very difficult problem, since it in particular implies $p_{n+1}-p_n 0$, in particular resolving Legendre's conjecture.\n\nGrimm proved that this is true if $k\\ll \\log n/\\log\\log n$. Erdős and Selfridge improved this to $k\\leq (1+o(1))\\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to\\[k\\ll\\left(\\frac{\\log n}{\\log\\log n}\\right)^3.\\]Laishram and Shorey [LaSh06] have verified this is true, for all $k\\geq 1$, for all $n\\leq 1.9\\times 10^{10}$. \n\nThis is problem B32 in Guy's collection [Gu04].\n\nSee also [860].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 January 2026. View history", "references": "#375: [Er72][Er73][ErGr80,p.71]" }, { "number": 376, "url": "https://www.erdosproblems.com/376", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients", "base representations" ], "oeis": [ "A030979" ], "formalized": "yes", "statement": "Are there infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $105$?", "commentary": "Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $pq$. \n\nThis is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ in base $5$, and digits $0,1,2,3$ in base $7$.\n\nThe sequence of such $n$ is A030979 in the OEIS.\n\nThe best result in this direction is due to Bloom and Croot [BlCr25], who proved that, if $p_1,p_2,p_3$ are sufficiently large primes, then there are infinitely many $n$ such that almost all of the base $p_i$ digits are $0$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $p_1p_2p_3$, except for a factor of size $\\leq n^\\epsilon$.\n\nThis is mentioned in problem B33 of Guy's collection [Gu04]. It is also discussed in an article of Pomerance [Po15c]. \n\nGraham offered \\$1000 for a solution to this problem (as mentioned in [Gu04] and [BeHa98]). \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#376: [ErGr80,p.71]" }, { "number": 377, "url": "https://www.erdosproblems.com/377", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [], "formalized": "yes", "statement": "Is there some absolute constant $C>0$ such that\\[\\sum_{p\\leq n}1_{p\\nmid \\binom{2n}{n}}\\frac{1}{p}\\leq C\\]for all $n$ (where the summation is restricted to primes $p\\leq n$)?", "commentary": "A question of Erdős, Graham, Ruzsa, and Straus [EGRS75], who proved that if $f(n)$ is the sum in question then\\[\\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n) = \\sum_{k=2}^\\infty \\frac{\\log k}{2^k}=\\gamma_0\\]and\\[\\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n)^2 = \\gamma_0^2,\\]so that for almost all integers $f(m)=\\gamma_0+o(1)$. They also prove that, for all large $n$,\\[f(n) \\leq c\\log\\log n\\]for some constant $c<1$. (It is trivial from Mertens estimates that $f(n)\\leq (1+o(1))\\log\\log n$.)\n\nA positive answer would imply that\\[\\sum_{p\\leq n}1_{p\\mid \\binom{2n}{n}}\\frac{1}{p}=(1-o(1))\\log\\log n,\\]and Erdős, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true.\n\nThis is mentioned in problem B33 of Guy's collection [Gu04]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#377: [EGRS75][Er79][ErGr80]" }, { "number": 382, "url": "https://www.erdosproblems.com/382", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A388850" ], "formalized": "no", "statement": "Let $u\\leq v$ be such that the largest prime dividing $\\prod_{u\\leq m\\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?", "commentary": "Erdős and Graham report it follows from results of Ramachandra that $v-u\\leq v^{1/2+o(1)}$. \n\nCambie has observed that the first question boils down to some old conjectures on prime gaps.\nBy Cramér's conjecture, for every $\\epsilon>0,$ for every $u$ sufficiently large there is a prime between $u$ and $u+u^\\epsilon$.\nThus for $u+u^\\epsilon0$. For any fixed $k$, there is therefore a positive 'probability' that there are $k$ consecutive integers around $q^2$ (for a prime $q$) all of whose prime divisors are bounded above by $q$, when $v-u\\geq k$. See [383] for a conjecture along these lines. A similar argument applies if we replace multiplicity $2$ with multiplicity $r$, for any fixed $r\\geq 2$.\n\nSee also [380].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#382: [ErGr80]" }, { "number": 383, "url": "https://www.erdosproblems.com/383", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of\\[\\prod_{0\\leq i\\leq k}(p^2+i)\\]is $p$?", "commentary": "A positive answer to this would give an answer to the second part of [382]. Heuristically, the 'probability' that $n$ has no prime divisors $\\geq n^{1/2}$ is $1-\\log 2>0$, so standard heuristics predict the answer to this is yes.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#383: [ErGr80]" }, { "number": 385, "url": "https://www.erdosproblems.com/385", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A322292" ], "formalized": "yes", "statement": "Let\\[F(n) = \\max_{\\substack{mn$ for all sufficiently large $n$? Does $F(n)-n\\to \\infty$ as $n\\to\\infty$?", "commentary": "A question of Erdős, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\\sqrt{n}$ (note that it is trivially $\\leq n+\\sqrt{n}$).\n\nTao has discussed this problem in a blog post.\n\nSarosh Adenwalla has observed that the first question is equivalent to [430]. Indeed, if $n$ is large and $a_i$ is the sequence defined in the latter problem, then [430] implies that there is a composite $a_j$ such that $a_j-p(a_j)>n$ and hence $F(n)>n$.\n\nSee also [463].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#385: [Er79d,p.73][ErGr80,p.74]" }, { "number": 386, "url": "https://www.erdosproblems.com/386", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A280992" ], "formalized": "yes", "statement": "Let $2\\leq k\\leq n-2$. Can $\\binom{n}{k}$ be the product of consecutive primes infinitely often? For example\\[\\binom{21}{2}=2\\cdot 3\\cdot 5\\cdot 7.\\]", "commentary": "Erdős and Graham write that 'a proof that this cannot happen infinitely often for $\\binom{n}{2}$ seems hopeless; probably this can never happen for $\\binom{n}{k}$ if $3\\leq k\\leq n-3$.'\n\nWeisenberg has provided four easy examples that show Erdős and Graham were too optimistic here:\\[\\binom{7}{3}=5\\cdot 7,\\]\\[\\binom{10}{4}= 2\\cdot 3\\cdot 5\\cdot 7,\\]\\[\\binom{14}{4} = 7\\cdot 11\\cdot 13,\\]and\\[\\binom{15}{6}=5\\cdot 7\\cdot 11\\cdot 13.\\]The known values of $n$ for which $\\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#386: [ErGr80,p.74]" }, { "number": 387, "url": "https://www.erdosproblems.com/387", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [], "formalized": "yes", "statement": "Is there an absolute constant $c>0$ such that, for all $1\\leq k< n$, the binomial coefficient $\\binom{n}{k}$ has a divisor in $(cn,n]$?", "commentary": "Erdős once conjectured that $\\binom{n}{k}$ must always have a divisor in $(n-k,n]$, but this was disproved by Schinzel and Erdős [Sc58]. A counterexample is given by $n=99215$ and $k=15$. Schinzel conjectured (see problem B34 of [Gu04]) that, for all sufficiently large $k$ which are not prime powers, there exists an $n$ such that $\\binom{n}{k}$ is not divisible by any integer in $(n-k,n]$.\n\nIn [Er78g] Erdős thought that this question had a negative answer.\n\nIt is easy to see that $\\binom{n}{k}$ always has a divisor in $[n/k,n]$. \n\nFaulkner [Fa66] proved that, if $p$ is the least prime $>2k$ and $n\\geq p$, then $\\binom{n}{k}$ has a prime divisor $\\geq p$ (except $\\binom{9}{2}$ and $\\binom{10}{3}$).\n\nThis is discussed in problems B33 and B34 of Guy's collection [Gu04], who says that Erdős conjectured this is true for any $c<1$ (if $n$ is sufficiently large).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#387: [ErGr76b,p.351][Er78g,p.315][ErGr80,p.74]" }, { "number": 388, "url": "https://www.erdosproblems.com/388", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Can one classify all solutions of\\[\\prod_{1\\leq i\\leq k_1}(m_1+i)=\\prod_{1\\leq j\\leq k_2}(m_2+j)\\]where $k_1,k_2>3$ and $m_1+k_1\\leq m_2$? Are there only finitely many solutions?", "commentary": "More generally, if $k_1>2$ then for fixed $a$ and $b$\\[a\\prod_{1\\leq i\\leq k_1}(m_1+i)=b\\prod_{1\\leq j\\leq k_2}(m_2+j)\\]should have only a finite number of solutions. \n\nSee also [363], [686], and [931].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#388: [Er76d][ErGr80][Er92e]" }, { "number": 389, "url": "https://www.erdosproblems.com/389", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A375071" ], "formalized": "yes", "statement": "Is it true that for every $n\\geq 1$ there is a $k$ such that\\[n(n+1)\\cdots(n+k-1)\\mid (n+k)\\cdots (n+2k-1)?\\]", "commentary": "Asked by Erdős and Straus. \nFor example when $n=2$ we have $k=5$:\\[2\\times 3 \\times 4 \\times 5\\times 6 \\mid 7 \\times 8 \\times 9\\times 10\\times 11.\\]and when $n=3$ we have $k=4$:\\[3\\times 4\\times 5\\times 6 \\mid 7\\times 8\\times 9\\times 10.\\]Bhavik Mehta has computed the minimal such $k$ for $1\\leq n\\leq 18$ (now available as A375071 on the OEIS).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#389: [ErGr80,p.75]" }, { "number": 390, "url": "https://www.erdosproblems.com/390", "status": "open", "prize": "no", "tags": [ "number theory", "factorials" ], "oeis": [ "A193429" ], "formalized": "yes", "statement": "Let $f(n)$ be the minimal $m$ such that\\[n! = a_1\\cdots a_k\\]with $n< a_1<\\cdots 0$?Is it true that, for $k\\geq 2$,\\[\\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)?\\]", "commentary": "In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős and Hall [ErHa78], who proved that in fact\\[\\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2.\\]Erdős and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.\n\nSince $t_2(p)=p-1$ for prime $p$ it is trivial that\\[\\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}.\\]Erdős and Hall [ErHa78] also note that $t_{n-1}(n!)=2$ and $t_{n-2}(n!)\\ll n$, which $n=2^r$ shows is the best possible. They ask about the behaviour of $t_{n-3}(n!)$ and also ask ask whether, for infinitely many $n$,\\[t_k(n!)< t_{k-1}(n!)-1\\]for all $1\\leq k0$ such that there are infinitely many $n$ where $m+\\epsilon \\omega(m)\\leq n$ for all $m\\phi(m+2)>\\cdots \\phi(m+k)?\\]Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "commentary": "Erdős [Er36b] proved that\\[F(n)\\asymp \\log\\log\\log n,\\]and similarly if we replace $\\phi$ with $\\sigma$ or $\\tau$ or $\\nu$ or any 'decent' additive or multiplicative function.\n\nWeisenberg has observed that the same questions could be asked for ordering patterns which allow equality (indeed, the final problem only makes sense if we allow equality).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#415: [ErGr80,p.82]" }, { "number": 416, "url": "https://www.erdosproblems.com/416", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A264810" ], "formalized": "yes", "statement": "Let $V(x)$ count the number of $n\\leq x$ such that $\\phi(m)=n$ is solvable. Does $V(2x)/V(x)\\to 2$? Is there an asymptotic formula for $V(x)$?", "commentary": "Pillai [Pi29] proved $V(x)=o(x)$. Erdős [Er35b] proved $V(x)=x(\\log x)^{-1+o(1)}$. \n\nThe behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance [MaPo88] proved\\[V(x)=\\frac{x}{\\log x}e^{(C+o(1))(\\log\\log\\log x)^2},\\]for some explicit constant $C>0$. Ford [Fo98] improved this to\\[V(x)\\asymp\\frac{x}{\\log x}e^{C_1(\\log\\log\\log x-\\log\\log\\log\\log x)^2+C_2\\log\\log\\log x-C_3\\log\\log\\log\\log x}\\]for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\\to 2$.\n\nIn [Er79e] Erdős asks further to estimate the number of $n\\leq x$ such that the smallest solution to $\\phi(m)=n$ satisfies $kx1$?", "commentary": "It is trivial that $V'(x) \\leq V(x)$. In [Er98] Erdős suggests the limit may be infinite. See also [416].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#417: [Er79e][ErGr80][Er98]" }, { "number": 420, "url": "https://www.erdosproblems.com/420", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "If $\\tau(n)$ counts the number of divisors of $n$ then let\\[F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}.\\]Is it true that\\[\\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty\\]for large $C$? Is it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$? More generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?", "commentary": "Erdős and Graham write that it is easy to show that $\\lim F(n^{1/2},n)=\\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.\n\nErdős, Graham, Ivić, and Pomerance [EGIP96] have proved that\\[\\liminf F(c\\log n, n) = 1\\]for any $c>0$, and\\[\\lim F(n^{4/9},n)=\\infty.\\](The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\\log n)^2)$, then for almost all $n$\\[F(f,n)\\sim 1.\\]van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies\\[\\limsup_{n\\to \\infty}F(g(n),n)=\\infty\\]for any $g(n)\\to \\infty$, and that Cramér's conjecture implies\\[\\lim F(g(n)(\\log n)^2, n)=\\infty\\]for any $g(n)\\to \\infty$>\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 December 2025. View history", "references": "#420: [ErGr80,p.83]" }, { "number": 421, "url": "https://www.erdosproblems.com/421", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A389544", "A390848" ], "formalized": "yes", "statement": "Is there a sequence $1\\leq d_11/e-\\epsilon$ for any $\\epsilon>0$.\n\nSee also [786].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#421: [ErGr80,p.84]" }, { "number": 422, "url": "https://www.erdosproblems.com/422", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A005185" ], "formalized": "yes", "statement": "Let $f(1)=f(2)=1$ and for $n>2$\\[f(n) = f(n-f(n-1))+f(n-f(n-2)).\\]Does $f(n)$ miss infinitely many integers? What is its behaviour?", "commentary": "Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#422: [ErGr80]" }, { "number": 423, "url": "https://www.erdosproblems.com/423", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A005243" ], "formalized": "no", "statement": "Let $a_1=1$ and $a_2=2$ and for $k\\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?", "commentary": "Asked by Hofstadter (in [Er77c] Erdős says Hofstadter was inspired by a similar question of Ulam). The sequence begins\\[1,2,3,5,6,8,10,11,\\ldots\\]and is A005243 in the OEIS.\n\nBolan and Tang [Ta26] have independently proved that there are infinitely many integers which do not appear in this sequence. In fact, the sequence $a_n-n$ is nondecreasing and unbounded.\n\nTang [Ta26] further proved that $a_n \\ll n^{\\frac{1}{c-1}+o(1)}$, where $c$ is any exponent such that $\\lvert A-A\\rvert \\geq \\lvert A\\rvert^{c-o(1)}$ for all convex sets $A$. In particular, the latest bounds of Cushman [Cu25] yield\\[a_n \\ll n^{\\frac{688}{413}+o(1)}\\leq n^{1.6659+o(1)},\\]and the conjecture of Erdős and Hegyvári that one can take $c=2$ in the convex set problem would imply $a_n\\leq n^{1+o(1)}$. It seems likely that $a_n=n+o(n)$ is the truth.\n\nTang has further proved an improved lower bound of\\[a_n=n+\\Omega(\\log\\log n).\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#423: [Er77c,p.71][ErGr80,p.83]" }, { "number": 424, "url": "https://www.erdosproblems.com/424", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A005244" ], "formalized": "yes", "statement": "Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\\neq j$. Is it true that the set of integers which eventually appear has positive density?", "commentary": "Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\\ldots$ and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory.\n\nIn [ErGr80] (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, but the answer to this is trivially no (as pointed out to me by Steinerberger): no integer $\\equiv 1\\pmod{3}$ is ever in the sequence, so the set of integers which appear has density at most $2/3$. This is easily seen by induction, and the fact that if $a,b\\in \\{0,2\\}\\pmod{3}$ then $ab-1\\in \\{0,2\\}\\pmod{3}$.\n\nPresumably it is the weaker question of whether a positive density of integers appear (as correctly asked in [Er77c]) that was also intended in [ErGr80]. As with many of Erdős' questions, by 'positive density' he most likely meant 'positive lower density' - in other words, does there exist $c>0$ such that for all large $x$ the number of values of $a_i$ in $[1,x]$ is at least $cx$?\n\nSee also Problem 63 of Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 March 2026. View history", "references": "#424: [Er77c,p.71][ErGr80,p.84]" }, { "number": 425, "url": "https://www.erdosproblems.com/425", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets" ], "oeis": [], "formalized": "no", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,N\\}$ such that the products $ab$ are distinct for all $a0?\\]", "commentary": "Erdős and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\\liminf$ by $\\limsup$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#428: [ErGr80]" }, { "number": 430, "url": "https://www.erdosproblems.com/430", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.Is it true that, for sufficiently large $n$, not all of this sequence can be prime?", "commentary": "Erdős and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.\n\nSarosh Adenwalla has observed that this problem is equivalent to (the first part of) [385]. Indeed, assuming a positive answer to that, for all large $n$, there exists a composite $mn-m$. It follows that such an $m$ is equal to some $a_i$ in the sequence defined for $[1,n)$, and $m$ is composite by assumption.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#430: [ErGr80]" }, { "number": 431, "url": "https://www.erdosproblems.com/431", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?", "commentary": "A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. (In [Er80] Erdős dates this to abou 25 years old, putting it at about 1955.)\n\nThe answer is surely no. The best result in this direction is due to Elsholtz and Harper [ElHa15], who showed that if $A,B$ are such sets then for all large $x$ we must have\\[\\frac{x^{1/2}}{\\log x\\log\\log x} \\ll \\lvert A \\cap [1,x]\\rvert \\ll x^{1/2}\\log\\log x\\]and similarly for $B$.\n\nElsholtz [El01] has proved there are no sets $A,B,C$ (all of size at least $2$) such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.\n\nGranville [Gr90] proved, conditional on the prime $k$-tuples conjecture, that there are infinite sets $B$ and $C$ such that\\[\\{ \\tfrac{b+c}{2}: b\\in B, c\\in C\\}\\]is a subset of the primes. Tao and Ziegler [TaZi23] gave an unconditional proof that there are infinite sets $B=\\{b_1<\\cdots\\}$ and $C=\\{c_1<\\cdots\\}$ such that\\[\\{ b_i+c_j : b_i\\in B, c_j\\in C, i1/2$, if $p$ is a sufficiently large prime then, for any $n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?", "commentary": "Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$. \n\nThis is discussed in this MathOverflow question.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#445: [ErGr80,p.89]" }, { "number": 450, "url": "https://www.erdosproblems.com/450", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [], "formalized": "no", "statement": "How large must $y=y(\\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?", "commentary": "It is not clear what the intended quantifier on $x$ is. Cambie has observed that if this is intended to hold for all $x$ then, provided\\[\\epsilon(\\log n)^\\delta (\\log\\log n)^{3/2}\\to \\infty\\]as $n\\to \\infty$, where $\\delta=0.086\\cdots$, there is no such $y$, which follows from an averaging argument and the work of Ford [Fo08]. \n\nOn the other hand, Cambie has observed that if $\\epsilon\\ll 1/n$ then $y(\\epsilon,n)\\sim 2n$: indeed, if $y<2n$ then this is impossible taking $x+n$ to be a multiple of the lowest common multiple of $\\{n+1,\\ldots,2n-1\\}$. On the other hand, for every fixed $\\delta\\in (0,1)$ and $n$ large every $2(1+\\delta)n$ consecutive elements contains many elements which are a multiple of an element in $(n,2n)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#450: [ErGr80,p.89]" }, { "number": 451, "url": "https://www.erdosproblems.com/451", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A386620" ], "formalized": "no", "statement": "Estimate $n_k$, the smallest integer $>2k$ such that $\\prod_{1\\leq i\\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.", "commentary": "Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'. \n\nIn [Er79d] Erdős writes that probably $n_kk^d$ for all constant $d$.\n\nAdenwalla observes that an easy upper bound is $n_k\\leq \\prod_{k\\log\\log n$ for all $n\\in I$?", "commentary": "Erdős [Er37] proved that the density of integers $n$ with $\\omega(n)>\\log\\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with\\[\\lvert I\\rvert \\geq (1+o(1))\\frac{\\log x}{(\\log\\log x)^2}.\\]It could be true that there is such an interval of length $(\\log x)^{k}$ for arbitrarily large $k$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 October 2025. View history", "references": "#452: [ErGr80,p.90]" }, { "number": 454, "url": "https://www.erdosproblems.com/454", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A389676", "A389677" ], "formalized": "yes", "statement": "Let\\[f(n) = \\min_{i0.352\\cdots.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 October 2025. View history", "references": "#455: [ErGr80,p.91]" }, { "number": 456, "url": "https://www.erdosproblems.com/456", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $p_n$ be the smallest prime $\\equiv 1\\pmod{n}$ and let $m_n$ be the smallest integer such that $n\\mid \\phi(m_n)$.Is it true that $m_na_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq in$ (this observation was made by Svyable using ChatGPT).\n\nUnfortunately, although both sources mention a forthcoming paper of Eggleton, Erdős, and Selfridge, I cannot find a candidate paper of theirs with this problem in, and hence the motivation behind this problem, and what the precise problem intended was, is unclear.\n\nChojecki has noted in the comments that a positive solution to the main problem would follow if\\[f(n) = \\sum_{aa}\\frac{1}{a}\\to \\infty,\\]where $P^-(\\cdot)$ is the least prime factor. Standard estimates on rough numbers show that $\\frac{1}{N}\\sum_{n\\leq N}f(n)\\gg \\log\\log N$, so $f(n)$ does diverge on average, but it is unclear whether $f(n)\\to \\infty$ for all $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 January 2026. View history", "references": "#460: [Er77c,p.64][ErGr80,p.91]" }, { "number": 461, "url": "https://www.erdosproblems.com/461", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p0$ such that\\[\\sum_{\\substack{n0$ such that\\[\\sum_{x\\leq n\\leq x+Cx^{1/2}(\\log x)^2}\\frac{p(n)}{n} \\gg 1\\]for all large $x$?", "commentary": "View the LaTeX source\n\n\n\n\n \n View history", "references": "#462: [ErGr80,p.92]" }, { "number": 463, "url": "https://www.erdosproblems.com/463", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "yes", "statement": "Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that\\[n+f(n)n}(m-p(m)),\\]and whether $n-F(n)\\sim cn^{1/2}$ for some $c>0$.\n\nSee also [385].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#463: [ErGr80][Er92e]" }, { "number": 467, "url": "https://www.erdosproblems.com/467", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\\leq x$ and a decomposition $\\{p\\leq x\\}=A\\sqcup B$ into two non-empty sets such that, for all $n0$, and Costa and Della Fiore [CoDe26] have further improved this to\\[t \\leq e^{c(\\log p)^{1/3}}.\\]\n The medium $A$ case: Pham and Sauermann [PhSa26] proved this, for any $0<\\alpha<1$, for all\\[ 1 \\ll_\\alpha t\\leq p^{1-\\alpha}.\\]\n The large $A$ case: Bedert, Bucić, Kravitz, Montgomery, and Müyesser [BBKMM25] proved this for all\\[p^{1-c}\\leq t\\leq (1-o(1))p\\]for some small constant $c>0$.\nThe very large $A$ case: Müyesser and Pokrovskiy [MuPo25] proved this for all\\[t\\geq (1-o(1))p.\\]\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 05 March 2026. View history", "references": "#475: [Er73][ErGr80]" }, { "number": 477, "url": "https://www.erdosproblems.com/477", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(k) : k\\in\\mathbb{Z}\\}$ such that $n=a+b$?", "commentary": "A question of Erdős and Graham, who thought the answer was negative. Clearly any such $A$ must be infinite.\n\nA simple proof that this is impossible for $f$ of degree $2$ is given in the comments (with the combined efforts of AlphaProof and Adenwalla): let $f(x)=c_2x^2+c_1x+c_0$ with $c_1\\neq 0$. For any infinite $A$ there exist $a,b\\in A$ such that $a-b=2c_1k$ for some $k\\geq 1$, whence\\[0\\neq a-b=f(k)-f(-k)\\]and so $n=a+f(-k)=b+f(k)$ has two distinct solutions. If $c_1=0$ then we can argue similarly finding $a,b\\in A$ with $a-b=4c_2k$ for some $k\\geq 1$, whence\\[0\\neq a-b=f(k+1)-f(k-1).\\]The same argument works for any $f$ such that $f(\\mathbb{Z})-f(\\mathbb{Z})$ contains all multiples of some fixed $q\\geq 1$, such as any polynomial of the shape\\[f(x)=g((x-k)^2)+c_1x+c_0\\]for some $g\\in \\mathbb{Z}[x]$ and $k,c_1,c_0\\in \\mathbb{Z}$ with $c_1\\neq 0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#477: [ErGr80,p.95]" }, { "number": 478, "url": "https://www.erdosproblems.com/478", "status": "open", "prize": "no", "tags": [ "number theory", "factorials" ], "oeis": [ "A210184" ], "formalized": "no", "statement": "Let $p$ be a prime and\\[A_p = \\{ k! \\pmod{p} : 1\\leq k1$, so the restriction $k\\neq 1$ is necessary. Erdős and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\\geq 1$, or if $k=-1$, but I cannot find such a paper. Tang has written a short note giving a proof for this case.\n\nAs an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\\equiv 3\\pmod{n}$ is $n=4700063497$. \n\nThe minimal such $n$ for each $k$ is A036236 in the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 December 2025. View history", "references": "#479: [ErGr80,p.96]" }, { "number": 483, "url": "https://www.erdosproblems.com/483", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics", "ramsey theory" ], "oeis": [ "A030126" ], "formalized": "no", "statement": "Let $f(k)$ be the minimal $N$ such that if $\\{1,\\ldots,N\\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?", "commentary": "The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are\\[(380)^{k/5}-O(1)\\leq f(k) \\leq (e-\\tfrac{1}{6}) k!.\\]Note that $380^{1/5}\\approx 3.2806$. The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik [ACPPRT21] (improving previous bounds of Exoo [Ex94] and Fredricksen and Sweet [FrSw00]). The upper bound is due to Xu, Xie, and Chen [XXC02] (improving previous bounds of Wan [Wa97] and Whitehead [Wh73]). See Eliahou [El19] for more on the upper bound.\n\nThe known values of $f$ are $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, and $f(5)=161$ (see A030126). (The equality $f(5)=161$ was established by Heule [He17]).\n\nSee also [183] (in particular a folklore observation gives $f(k)\\leq R(3;k)-1$).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#483: [Er61,p.233][Er65,p.188]" }, { "number": 486, "url": "https://www.erdosproblems.com/486", "status": "open", "prize": "no", "tags": [ "number theory", "primitive sets" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let\\[B = \\{ m\\in \\mathbb{N} : m\\not\\in X_n\\pmod{n}\\textrm{ for all }n\\in A\\textrm{ with }m>n\\}.\\]Must $B$ have a logarithmic density, i.e. is it true that\\[\\lim_{x\\to \\infty} \\frac{1}{\\log x}\\sum_{\\substack{m\\in B\\\\ mn\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]", "commentary": "The constant $2$ would be the best possible here, as witnessed by taking $A=\\{a\\}$, $n=2a-1$, and $m=2a$.\n\nThis problem is also discussed in problem E5 of Guy's collection [Gu04].\n\nIn [Er61] this problem is as stated above, but with $a\\mid n$ in the definition of $B$ replaced by $a\\nmid n$. This is most likely a typo (especially since the problem is also given as stated above in [Er66]). There have been several counterexamples given for this alternate problem. Cambie has observed that, if $A$ is the set of primes bounded above by $n$, and $m=2n$, then\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}=\\frac{\\pi(2n)-\\pi(n)+1}{2n}\\sim \\frac{1}{2\\log n}\\]while\\[\\frac{\\lvert B\\cap [1,n]\\rvert}{n}=\\frac{1}{n}.\\]Further concrete counterexamples, found by Alexeev and Aristotle, are given in the comments section.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#488: [Er61,p.236][Er66,p.150][Er80,p.112]" }, { "number": 489, "url": "https://www.erdosproblems.com/489", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let\\[B=\\{ n\\geq 1 : a\\nmid n\\textrm{ for all }a\\in A\\}.\\]If $B=\\{b_1393$, the points determine at least $\\binom{n-1}{2}$ distinct circles. There was an error, observed by Purdy and Smith [PuSm], who noted that Elliott's proof actually gives a lower bound of\\[\\binom{n-1}{2}+1-\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor,\\]again for all $n>393$. This is best possible, as witnessed by a circle with $n-1$ points and a single point off the circle. This corrected lower bound was also reported by Bálint and Bálintová [BaBa94], although without any explanation.\n\nThe problem appears to remain open for small $n$. Segre observed that projecting a cube onto a plane shows that the lower bound $\\binom{n-1}{2}$ is false for $n=8$.\n\nSee also [104] and [831].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#506: [Er61,p.245]" }, { "number": 507, "url": "https://www.erdosproblems.com/507", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "yes", "statement": "Let $\\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\\alpha(n)$. Estimate $\\alpha(n)$.", "commentary": "Heilbronn's triangle problem. It is trivial that $\\alpha(n) \\ll 1/n$. Erdős observed that $\\alpha(n)\\gg 1/n^2$. The current best bounds are\\[\\frac{\\log n}{n^2}\\ll \\alpha(n) \\ll \\frac{1}{n^{7/6+o(1)}}.\\]The lower bound is due to Komlós, Pintz, and Szemerédi [KPS82]. The upper bound is due to Cohen, Pohoata, and Zakharov [CPZ24] (improving on their earlier work [CPZ23] which itself improves an exponent of $8/7$ due to Komlós, Pintz, and Szemerédi [KPS81]).\n\nThis problem is Problem 77 on Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 December 2025. View history", "references": "#507: [Er61,p.246][Er75f,p.107]" }, { "number": 508, "url": "https://www.erdosproblems.com/508", "status": "open", "prize": "no", "tags": [ "geometry", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?", "commentary": "The Hadwiger-Nelson problem. Let $\\chi$ be the chromatic number of the plane. An equilateral triangle trivially shows that $\\chi\\geq 3$. There are several small graphs that show $\\chi\\geq 4$ (in particular the Moser spindle and Golomb graph). The best bounds currently known are\\[5 \\leq \\chi \\leq 7.\\]The lower bound is due to de Grey [dG18]. The upper bound can be seen by colouring the plane by tesselating by hexagons with diameter slightly less than $1$.\n\nMatolcsi, Ruzsa, Varga, and Zsámboki have proved that the fractional chromatic number of the plane is at least $4$. Croft [Cr67] has proved it is at most $4.359\\cdots$.\n\nSee also [704], [705], and [706]. The independence number of a finite unit distance graph is the topic of [1070].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#508: [Er61,p.246][Er75f,p.107][Er81]" }, { "number": 509, "url": "https://www.erdosproblems.com/509", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "Let $f(z)\\in\\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\\[\\{ z\\in \\mathbb{C} : \\lvert f(z)\\rvert \\leq 1\\}\\]be covered by a set of circles the sum of whose radii is $\\leq 2$?", "commentary": "Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (see [1046]).\n\nThe generalisation of this to higher dimensions was asked by Erdős as Problem 4.23 in [Ha74].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 December 2025. View history", "references": "#509: [Er61,p.246][Ha74]" }, { "number": 510, "url": "https://www.erdosproblems.com/510", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "If $A\\subset \\mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\\theta$ such that\\[\\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}?\\]", "commentary": "Chowla's cosine problem. Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), proved an upper bound of\\[-\\exp(O(\\sqrt{\\log N})).\\]Polynomial bounds were proved independently by Bedert [Be25c] and Jin, Milojević, Tomon, and Zhang [JMTZ25]. The best bound follows from the method of Bedert [Be25c], which proved the existence of some $c>0$ such that, for all $A$ of size $N$,\\[\\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/7}.\\]The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.\n\nThis problem is Problem 81 on Green's open problems list.\n\nThis is related to [256].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#510: [Er61,p.248]" }, { "number": 513, "url": "https://www.erdosproblems.com/513", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\\[\\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rvert}{\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert}?\\]", "commentary": "Let $B$ be the supremum of all such values. \n\nIt is trivial that $B\\in [1/2,1]$. Kövári (unpublished) observed that it must be $>1/2$. Gray and Shah [GrSh63] give an argument of Clunie which shows $B\\leq 2/\\pi$. Clunie and Hayman [ClHa64] improved both bounds to\\[4/7< B \\leq 2/\\pi-c\\]for some absolute constant $c>0$. He and Tang [HeTa26] have improved the lower bound to\\[B> 0.5850724\\](note that $4/7\\approx 0.57143$). This was further improved slightly to $0.5850788$ by GPT (as prompted by Sothanaphan). Note that $2/\\pi \\approx 0.63662$.\n\nSee also [227].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 April 2026. View history", "references": "#513: [Er61,p.249]" }, { "number": 514, "url": "https://www.erdosproblems.com/514", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,\\[\\lvert f(z)/z^n\\rvert \\to \\infty\\]as $z\\to \\infty$ along $L$?Can the length of this path be estimated in terms of $M(r)=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$? Does there exist a path along which $\\lvert f(z)\\rvert$ tends to $\\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\\epsilon$)?", "commentary": "Boas (unpublished) has proved the first part, that such a path must exist.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#514: [Er61,p.249][Er82e]" }, { "number": 517, "url": "https://www.erdosproblems.com/517", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "Let $f(z)=\\sum_{k=1}^\\infty a_kz^{n_k}$ be an entire function (with $a_k\\neq 0$ for all $k\\geq 1$). Is it true that if $n_k/k\\to \\infty$ then $f(z)$ assumes every value infinitely often?", "commentary": "A conjecture of Fejér and Pólya. \n\nFejér [Fe08] proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value at least once, and Biernacki [Bi28] proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value infinitely often.\n\nPólya [Po29] proved that if $f$ has finite order then $f(z)$ assumes every value infinitely often under the assumption that $\\limsup (n_{k+1}-n_k)=\\infty$.\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 29 December 2025. View history", "references": "#517: [Er61,p.250]" }, { "number": 520, "url": "https://www.erdosproblems.com/520", "status": "open", "prize": "no", "tags": [ "number theory", "probability" ], "oeis": [], "formalized": "yes", "statement": "Let $f$ be a Rademacher multiplicative function: a random $\\{-1,0,1\\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\\cdots p_r)=f(p_1)\\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely,\\[\\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N\\log\\log N}}=c?\\]", "commentary": "Note that if we drop the multiplicative assumption, and simply assign $f(m)=\\pm 1$ at random, then this statement is true (with $c=\\sqrt{2}$), the law of the iterated logarithm.\n\n Wintner [Wi44] proved that, almost surely,\\[\\sum_{m\\leq N}f(m)\\ll N^{1/2+o(1)},\\]and Erdős improved the right-hand side to $N^{1/2}(\\log N)^{O(1)}$. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely,\\[\\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{2+o(1)}.\\]Caich [Ca24b] has improved this to\\[\\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{3/4+o(1)}.\\]Harper [Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\\log\\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely\\[\\sum_{m\\leq N}f(m) \\ll N^{1/2}(\\log\\log N)^{1/4+o(1)}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#520: [Er61,p.251]" }, { "number": 521, "url": "https://www.erdosproblems.com/521", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials", "probability" ], "oeis": [], "formalized": "no", "statement": "Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from $\\{-1,1\\}$. If $R_n$ counts the number of real roots of $f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely,\\[\\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}?\\]", "commentary": "Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\n\nIt is ambiguous in [Er61] whether Erdős intended the coefficients to be uniformly chosen from $\\{-1,1\\}$ or $\\{0,1\\}$. In the latter case, the constant $\\frac{2}{\\pi}$ should be $\\frac{1}{\\pi}$ (see the discussion in the comments).\n\nIn the case of $\\{-1,1\\}$ Do [Do24] proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely,\\[\\lim_{n\\to \\infty}\\frac{R_n[-1,1]}{\\log n}=\\frac{1}{\\pi}.\\]See also [522].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#521: [Er61,p.252]" }, { "number": 522, "url": "https://www.erdosproblems.com/522", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials", "probability" ], "oeis": [], "formalized": "yes", "statement": "Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where $\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$. Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then\\[\\frac{R_n}{n/2}\\to 1\\]almost surely?", "commentary": "Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\\pm 1$ values. Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\n\nThere is some ambiguity whether Erdős intended the coefficients to be in $\\{-1,1\\}$ or $\\{0,1\\}$ - see the comments section.\n\nA weaker version of this was solved by Yakir [Ya21], who proved that\\[\\frac{R_n}{n/2}\\to 1\\]in probability. (This weaker claim was also asked by Erdős, and also appears in a book of Hayman [Ha67].) More precisely,\\[\\lim_{n\\to \\infty} \\mathbb{P}(\\lvert R_n-n/2\\rvert \\geq n^{9/10}) =0.\\]See also [521].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#522: [Er61,p.252]" }, { "number": 524, "url": "https://www.erdosproblems.com/524", "status": "open", "prize": "no", "tags": [ "analysis", "probability", "polynomials" ], "oeis": [], "formalized": "no", "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t\\in(0,1)$) for\\[M_n(t)=\\max_{x\\in [-1,1]}\\left\\lvert \\sum_{k\\leq n}(-1)^{\\epsilon_k(t)}x^k\\right\\rvert?\\]", "commentary": "A problem of Salem and Zygmund [SaZy54]. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that\\[M_n(t) \\ll \\left(\\frac{n}{\\log\\log n}\\right)^{1/2}.\\]Erdős (unpublished) showed that for almost all $t$ and every $\\epsilon>0$ we have $\\lim_{n\\to \\infty}M_n(t)/n^{1/2-\\epsilon}=\\infty$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#524: [Er61,p.253]" }, { "number": 528, "url": "https://www.erdosproblems.com/528", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [ "A387897", "A156816" ], "formalized": "no", "statement": "Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine\\[C_k=\\lim_{n\\to\\infty}f(n,k)^{1/n}.\\]", "commentary": "The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton [HM54] showed that this limit exists, and it is trivial that $k\\leq C_k\\leq 2k-1$.\n\nKesten [Ke63] proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade [CLS07].\n\nConway and Guttmann [CG93] showed that $C_2\\geq 2.62$ and Alm [Al93] showed that $C_2\\leq 2.696$. Jacobsen, Scullard, and Guttmann [JSG16] have computed the first few decimal places of $C_2$, showing that\\[C_2 = 2.6381585303279\\cdots.\\]See also [529].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#528: [Er61,p.254]" }, { "number": 529, "url": "https://www.erdosproblems.com/529", "status": "open", "prize": "no", "tags": [ "geometry", "probability" ], "oeis": [], "formalized": "no", "statement": "Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that\\[\\lim_{n\\to \\infty}\\frac{d_2(n)}{n^{1/2}}= \\infty?\\]Is it true that\\[d_k(n)\\ll n^{1/2}\\]for $k\\geq 3$?", "commentary": "Slade [Sl87] proved that, for $k$ sufficiently large, $d_k(n)\\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade ([HaSl91] and [HaSl92]) proved this for all $k\\geq 5$.\n\nFor $k=2$ Duminil-Copin and Hammond [DuHa13] have proved that $d_2(n)=o(n)$.\n\nIt is now conjectured that $d_k(n)\\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of [MaSl93]) that $d_2(n)\\sim Dn^{3/4}$, $d_3(n)\\sim n^{\\nu}$ where $\\nu\\approx 0.59$, and $d_4(n)\\sim D(\\log n)^{1/8}n^{1/2}$.\n\nMadras and Slade [MaSl93] have a monograph on the topic of self-avoiding walks.\n\nSee also [528].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#529: [Er61,p.254]" }, { "number": 530, "url": "https://www.erdosproblems.com/530", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets" ], "oeis": [ "A143824" ], "formalized": "no", "statement": "Let $\\ell(N)$ be maximal such that in any finite set $A\\subset \\mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\\ell(N)$.In particular, is it true that $\\ell(N)\\sim N^{1/2}$?", "commentary": "Originally asked by Riddell [Ri69]. Erdős noted the bounds\\[N^{1/3} \\ll \\ell(N) \\leq (1+o(1))N^{1/2}\\](the upper bound following from the case $A=\\{1,\\ldots,N\\}$). The lower bound was improved to $N^{1/2}\\ll \\ell(N)$ by Komlós, Sulyok, and Szemerédi [KSS75]. The correct constant is unknown, but it is likely that the upper bound is true, so that $\\ell(N)\\sim N^{1/2}$.\n\nIn [AlEr85] Alon and Erdős make the stronger conjecture that perhaps $A$ can always be written as the union of at most $(1+o(1))N^{1/2}$ many Sidon sets. (This is easily verified for $A=\\{1,\\ldots,N\\}$ using standard constructions of Sidon sets.)\n\nThis is discussed in problem C9 of Guy's collection [Gu04].\n\nSee also [1088] and [1208] for a higher-dimensional generalisation.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#530: [Er73,p.120][Er75f,p.104][Er80,p.109][Er80e]" }, { "number": 531, "url": "https://www.erdosproblems.com/531", "status": "open", "prize": "no", "tags": [ "number theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $F(k)$ be the minimal $N$ such that if we two-colour $\\{1,\\ldots,N\\}$ there is a set $A$ of size $k$ such that all subset sums $\\sum_{a\\in S}a$ (for $\\emptyset\\neq S\\subseteq A$) are monochromatic. Estimate $F(k)$.", "commentary": "The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.\n\nErdős and Spencer [ErSp89] proved that\\[F(k) \\geq 2^{ck^2/\\log k}\\]for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner [BENTW17] have improved this to\\[F(k) \\geq 2^{2^{k-1}/k}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#531: [Er73]" }, { "number": 535, "url": "https://www.erdosproblems.com/535", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.", "commentary": "Erdős [Er64] proved that\\[f_r(N) \\leq N^{\\frac{3}{4}+o(1)},\\]and Abbott and Hanson [AbHa70] improved this exponent to $1/2$. Erdős [Er64] proved the lower bound\\[f_3(N) > N^{\\frac{c}{\\log\\log N}}\\]for some constant $c>0$, and conjectured this should also be an upper bound.\n\nErdős writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that $\\omega(n)=k$ for all $n\\in A$ and there does not exist $a_1,\\ldots,a_r\\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\\neq j$ and $(a_i/d,d)=1$ for all $i$. (As Cong points out in the comments there is likely an assumption that $A$ consists of squarefree integers.)\n\nThe conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erdős and Rado gives the upper bound $c_r^kk!$.\n\nSee also [536].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#535: [Er69][Er70][Er73]" }, { "number": 536, "url": "https://www.erdosproblems.com/536", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple?", "commentary": "This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erdős [Er62] (with a proof given in [Er70]).\n\nThis was also asked by Erdős at the 1991 problem session of West Coast Number Theory.\n\nIn the comments Weisenberg sketches a construction of a set $A\\subseteq [1,N]$ without this property such that\\[\\lvert A\\rvert \\gg (\\log\\log N)^{f(N)}\\frac{N}{\\log N}\\]for some $f(N)\\to \\infty$. Weisenberg also sketches a proof of the main problem when $\\epsilon>\\frac{221}{225}$.\n\nSee also [535], [537], and [856]. A related combinatorial problem is asked at [857].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 12 January 2026. View history", "references": "#536: [Er64,p.646][Er70,p.124][Er73,p.124]" }, { "number": 538, "url": "https://www.erdosproblems.com/538", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$. Give the best possible upper bound for\\[\\sum_{n\\in A}\\frac{1}{n}.\\]", "commentary": "Erdős observed that\\[\\sum_{n\\in A}\\frac{1}{n}\\sum_{p\\leq N}\\frac{1}{p}\\leq r\\sum_{m\\leq N^2}\\frac{1}{m}\\ll r\\log N,\\]and hence\\[\\sum_{n\\in A}\\frac{1}{n} \\ll r\\frac{\\log N}{\\log\\log N}.\\]See also [536] and [537].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#538: [Er73]" }, { "number": 539, "url": "https://www.erdosproblems.com/539", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set\\[\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}\\]has size at least $h(n)$. Estimate $h(n)$.", "commentary": "Erdős and Szemerédi proved that\\[n^{1/2} \\ll h(n) \\ll n^{1-c}\\]for some constant $c>0$. The upper bound has been improved to $h(n)\\ll n^{2/3}$ by Freiman and Lev. A proof of these bounds can be found in a paper of Granville and Roesler [GrRo99].\n\nGranville and Roesler also reformulate this problem as one in combinatorial geometry: given $A\\subseteq \\mathbb{Z}^d$ of size $n$ what is the minimum size possible of\\[\\{ \\delta(\\mathbf{a},\\mathbf{b}) : \\mathbf{a},\\mathbf{b}\\in A\\}\\]where $\\delta(\\mathbf{a},\\mathbf{b})$ has $i$th coordinate $\\max(0,a_i-b_i)$. Using this formulation that obtain improved lower bounds in fixed small dimension $d$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#539: [Er73]" }, { "number": 544, "url": "https://www.erdosproblems.com/544", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A000791" ], "formalized": "no", "statement": "Show that\\[R(3,k+1)-R(3,k)\\to\\infty\\]as $k\\to \\infty$. Similarly, prove or disprove that\\[R(3,k+1)-R(3,k)=o(k).\\]", "commentary": "A problem of Erdős and Sós.\n\nThis problem is #8 in Ramsey Theory in the graphs problem collection.\n\nSee also [165] and [1014].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#544: [Er81c][Er93,p.339]" }, { "number": 545, "url": "https://www.erdosproblems.com/545", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442" ], "formalized": "no", "statement": "Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t0$.\nThis problem is #14 in Ramsey Theory in the graphs problem collection.\n\nSee also [549].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#547: [BuEr76]" }, { "number": 548, "url": "https://www.erdosproblems.com/548", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $n\\geq k+1$. Every graph on $n$ vertices with at least $\\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.", "commentary": "A problem of Erdős and Sós, who also conjectured that every graph with at least\\[\\max\\left( \\binom{2k-1}{2}+1, (k-1)n-(k-1)^2+\\binom{k-1}{2}+1\\right)\\]many edges contains every forest with $k$ edges. (Erdős and Gallai [ErGa59] proved that this is the threshold which guarantees containing $k$ independent edges.)\n\nIn [Er78] Erdős says this is trivial for a star, and he and Gallai had proved it for a path. \n\nIt can be easily proved by induction that every graph on $n$ vertices with at least $n(k-1)+1$ edges contains every tree on $k+1$ vertices.\n\nBrandt and Dobson [BrDo96] have proved this for graphs of girth at least $5$. Wang, Li, and Liu [WLL00] have proved this for graphs whose complements have girth at least $5$. Saclé and Woznik [SaWo97] have proved this for graphs which contain no cycles of length $4$. Yi and Li [YiLi04] have proved this for graphs whose complements contain no cycles of length $4$.\n\nImplies [547] and [557].\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#548: [Er64c][Er74c,p.78][Er78,p.30][Er93,p.345][Va99,3.55]" }, { "number": 550, "url": "https://www.erdosproblems.com/550", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\\ldots,m_k$ then prove that\\[R(T,G)\\leq (\\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.\\]", "commentary": "Chvátal [Ch77] proved that $R(T,K_m)=(m-1)(n-1)+1$.\n\nThis problem is #16 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#550: [EFRS85]" }, { "number": 551, "url": "https://www.erdosproblems.com/551", "status": "decidable", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Prove that\\[R(C_k,K_n)=(k-1)(n-1)+1\\]for $k\\geq n\\geq 3$ (except when $n=k=3$).", "commentary": "Asked by Erdős, Faudree, Rousseau, and Schelp, who also ask for the smallest value of $k$ such that this identity holds (for fixed $n$). They also ask, for fixed $n$, what is the minimum value of $R(C_k,K_n)$?\n\nThis identity was proved for $k>n^2-2$ by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to $k\\geq 4n+2$.\n\nKeevash, Long, and Skokan [KLS21] have proved this identity when $k\\geq C\\frac{\\log n}{\\log\\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$.\n\n\nThis problem is #18 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#551: [EFRS78]" }, { "number": 552, "url": "https://www.erdosproblems.com/552", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A006672" ], "formalized": "no", "statement": "Determine the Ramsey number\\[R(C_4,S_n),\\]where $S_n=K_{1,n}$ is the star on $n+1$ vertices.In particular, is it true that, for any $c>0$, there are infinitely many $n$ such that\\[R(C_4,S_n)\\leq n+\\sqrt{n}-c?\\]", "commentary": "A problem of Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS89]. Erdős often asked about $R(C_4,S_n)$ in the equivalent formulation of asking for a bound on the minimum degree of a graph which would guarantee the existence of a $C_4$ (see [85]).\n\nIt is known that\\[ n+\\sqrt{n}-6n^{11/40} \\leq R(C_4,S_n)\\leq n+\\lceil\\sqrt{n}\\rceil+1.\\]The lower bound is due to [BEFRS89], the upper bound is due to Parsons [Pa75]. The lower bound of [BEFRS89] is related to gaps between primes, and assuming e.g. Cramer's conjecture on gaps between primes their lower bound would be $n+\\sqrt{n}-n^{o(1)}$.\n\nErdős offered \\$100 for a proof or disproof of the second question in [BEFRS89]. In [Er96] Erdős asks (an equivalent formulation of) whether $R(C_4,S_n)\\geq n+\\sqrt{n}-O(1)$, but says this is probably 'too optimistic'.\n\nThey also ask, if $f(n)=R(C_4,S_n)$, whether $f(n+1)=f(n)$ infinitely often, and is the density of such $n$ $0$? Also, is it true that $f(n+1)\\leq f(n)+2$ for all $n$? A similar question about an equivalent function is the subject of [85].\n\nParsons [Pa75] proved that\\[R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil\\]whenever $n=q^2+1$ for a prime power $q$ and\\[R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+1\\]whenever $n=q^2$ for a prime power $q$ (in particular both equalities occur infinitely often).\n\nThis has been extended in various works, all in the cases $n=q^2\\pm t$ for some $0\\leq t\\leq q$ and prime power $q$. We refer to the work of Parsons [Pa75], Wu, Sun, Zhang, and Radziszowski [WSZR15], and Zhang, Chen, and Cheng ([ZCC17] and [ZCC17b]) for a precise description. In every known case\\[R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\},\\]and Zhang, Chen, and Cheng [ZCC17] speculate whether this is in fact true for all $n\\geq 2$ (whence the answer to the question above would be no).\n\nThis problem is #19 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#552: [BEFRS89][Er93,p.345][Er94b][Er95][Er96]" }, { "number": 554, "url": "https://www.erdosproblems.com/554", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that\\[\\lim_{k\\to \\infty}\\frac{R_k(C_{2n+1})}{R_k(K_3)}=0\\]for any $n\\geq 2$.", "commentary": "A problem of Erdős and Graham. The problem is open even for $n=2$.\n\nSchur [Sc16] proved\\[C^k \\ll R_k(K_3) \\ll k!\\]for some constant $C>0$. Erdős conjectured (see [183]) that $R_k(K_3) \\leq C^k$ for some contant $C>0$.\n\nBondy and Erdős [BoEr73] and Erdős and Graham [ErGr75] proved\\[n2^k+1\\leq R_k(C_{2n+1})\\leq 2n(k+2)!.\\]The lower bound is sharp for fixed $k$ and large enough $n$, as shown by Jenssen and Skokan [JeSk21]. Day and Johnson [DaJo17] have proved that, for fixed $n$,\\[R_k(C_{2n+1})\\geq 2n(2+c_n)^{k-1}\\]for all large $k$, for some constant $c_n>0$.\n\nAxenovich, Cames van Batenburg, Janzer, Michel, and Rundström [ACJMR25] have improved the upper bound to\\[R_k(C_{2n+1}) \\leq (4n-2)^k k^{k/n}+1.\\]In particular, there exists some absolute constant $c>0$ such that\\[R_k(C_{2n+1})\\leq (Cn)^kk!^{1/n}.\\]This problem is #23 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#554: [Er81c]" }, { "number": 555, "url": "https://www.erdosproblems.com/555", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of\\[R_k(C_{2n}).\\]", "commentary": "A problem of Erdős and Graham. Erdős [Er81c] gives the bounds\\[k^{1+\\frac{1}{2n}}\\ll R_k(C_{2n})\\ll k^{1+\\frac{1}{n-1}}.\\]Chung and Graham [ChGr75] showed that\\[R_k(C_4)>k^2-k+1\\]when $k-1$ is a prime power and\\[R_k(C_4)\\leq k^2+k+1\\]for all $k$.\n\nThis problem is #24 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#555: [Er81c]" }, { "number": 556, "url": "https://www.erdosproblems.com/556", "status": "decidable", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A389335" ], "formalized": "no", "statement": "Let $R_3(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $3$-coloured then there must be a monochromatic copy of $G$. Show that\\[R_3(C_n) \\leq 4n-3.\\]", "commentary": "A problem of Bondy and Erdős. This inequality is best possible for odd $n$. \n\nLuczak [Lu99] has shown that $R_3(C_n)\\leq (4+o(1))n$ for all $n$, and in fact $R_3(C_n)\\leq 3n+o(n)$ for even $n$.\n\nKohayakawa, Simonovits, and Skokan [KSS05] proved this conjecture when $n$ is sufficiently large and odd. Benevides and Skokan [BeSk09] proved that if $n$ is sufficiently large and even then $R_3(C_n)=2n$.\n\nThis problem is #25 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#556: [Er81][Er81c]" }, { "number": 557, "url": "https://www.erdosproblems.com/557", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that\\[R_k(T)\\leq kn+O(1)\\]for any tree $T$ on $n$ vertices?", "commentary": "A problem of Erdős and Graham. Implied by [548].\n\nThis would be best possible since, for example, $R_k(S_n)\\geq kn-O(k)$ if $S_n=K_{1,n-1}$ is a star on $n$ vertices.\n\nThis problem is #26 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#557: [ErGr75,p.516]" }, { "number": 558, "url": "https://www.erdosproblems.com/558", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine\\[R_k(K_{s,t})\\]where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.", "commentary": "Chung and Graham [ChGr75] prove the general bounds\\[(2\\pi\\sqrt{st})^{\\frac{1}{s+t}}\\left(\\frac{s+t}{e^2}\\right)k^{\\frac{st-1}{s+t}}\\leq R_k(K_{s,t})\\leq (t-1)(k+k^{1/s})^s\\]and determined\\[R_k(K_{2,2})=(1+o(1))k^2.\\]Alon, Rónyai, and Szabó [ARS99] have proved that\\[R_k(K_{3,3})=(1+o(1))k^3\\]and that if $s\\geq (t-1)!+1$ then\\[R_k(K_{s,t})\\asymp k^t.\\]This problem is #27 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#558: [Er81c]" }, { "number": 560, "url": "https://www.erdosproblems.com/560", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. Determine\\[\\hat{R}(K_{n,n}),\\]where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.", "commentary": "We know that\\[\\frac{1}{60}n^22^n<\\hat{R}(K_{n,n})< \\frac{3}{2}n^32^n.\\]The lower bound (which holds for $n\\geq 6$) was proved by Erdős and Rousseau [ErRo93]. The upper bound was proved by Erdős, Faudree, Rousseau, and Schelp [EFRS78b] and Nešetřil and Rödl [NeRo78].\n\nConlon, Fox, and Wigderson [CFW23] have proved that, for any $s\\leq t$,\\[\\hat{R}(K_{s,t})\\gg s^{2-\\frac{s}{t}}t2^s,\\]and prove that when $t\\gg s\\log s$ we have $\\hat{R}(K_{s,t})\\asymp s^2t2^s$. They conjecture that this should hold for all $s\\leq t$, and so in particular we should have $\\hat{R}(K_{n,n})\\asymp n^32^n$. \n\nThis problem is #29 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#560: [EFRS82]" }, { "number": 561, "url": "https://www.erdosproblems.com/561", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\\cup_{i\\leq s} K_{1,n_i}$ and $F_2=\\cup_{j\\leq t} K_{1,m_j}$ with $n_1\\geq \\cdots \\geq n_s\\geq 1$ and $m_1\\geq \\cdots \\geq m_t\\geq 1$. Prove that\\[\\hat{R}(F_1,F_2) = \\sum_{2\\leq k\\leq s+t}l_k\\]where\\[l_k=\\max\\{n_i+m_j-1 : i+j=k\\}.\\]", "commentary": "Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS78] proved this when all the $n_i$ are identical and all the $m_i$ are identical.\n\nGyőri and Schelp [GySc02] proved this when $\\binom{l_k}{2}> \\sum_{k\\leq i\\leq s+t}l_i$ for all $2\\leq k\\leq s+t$. More special cases, including all cases when $s=1$, or $s=2$ and $n_1=n_2$, or all $n_i$ and $m_j$ are odd, or all $n_i$ are an identical odd number and $m_1$ is odd, were proved by Davoodi, Javadi, Kamranian, and Raeisi [DJKR25]. \n\nThis problem is #30 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#561: [BEFRS78]" }, { "number": 562, "url": "https://www.erdosproblems.com/562", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory", "hypergraphs" ], "oeis": [], "formalized": "yes", "statement": "Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.Prove that, for $r\\geq 3$,\\[\\log_{r-1} R_r(n) \\asymp_r n,\\]where $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like\\[2^{2^{\\cdots n}}\\]where the tower of exponentials has height $r-1$?", "commentary": "A problem of Erdős, Hajnal, and Rado [EHR65]. A generalisation of [564].\n\nThis problem is #38 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#562: [EHR65]" }, { "number": 563, "url": "https://www.erdosproblems.com/563", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour. Prove that, for every $0\\leq \\alpha< 1/2$,\\[F(n,\\alpha)\\sim c_\\alpha\\log n\\]for some constant $c_\\alpha$ depending only on $\\alpha$.", "commentary": "It is easy to show via the probabilistic method that, for every $0\\leq \\alpha<1/2$,\\[F(n,\\alpha)\\asymp_\\alpha \\log n.\\]Note that when $\\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.\n\nSee also [161] for a generalisation to hypergraphs.\n\nThis problem is #39 in Ramsey Theory in the graphs problem collection.\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#563: [Er90b,p.21]" }, { "number": 564, "url": "https://www.erdosproblems.com/564", "status": "open", "prize": "$500", "tags": [ "graph theory", "ramsey theory", "hypergraphs" ], "oeis": [], "formalized": "yes", "statement": "Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.Is there some constant $c>0$ such that\\[R_3(n) \\geq 2^{2^{cn}}?\\]", "commentary": "A special case of [562]. A problem of Erdős, Hajnal, and Rado [EHR65], who prove the bounds\\[2^{cn^2}< R_3(n)< 2^{2^{n}}\\]for some constant $c>0$.\n\nErdős, Hajnal, Máté, and Rado [EHMR84] have proved a doubly exponential lower bound for the corresponding problem with $4$ colours. \n\nThis problem is #37 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#564: [EHR65][Er81][Er97c]" }, { "number": 566, "url": "https://www.erdosproblems.com/566", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then\\[R(G,H)\\ll m?\\]", "commentary": "In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erdős, Faudree, Rousseau, and Schelp [EFRS93] have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.\n\nImplies [567].\n\nThis problem is #31 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#566: [EFRS93]" }, { "number": 567, "url": "https://www.erdosproblems.com/567", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then\\[R(G,H)\\ll m?\\]", "commentary": "In other words, is $G$ Ramsey size linear? A special case of [566]. In [Er95] Erdős specifically asks about the case $G=K_{3,3}$.\n\nThe graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\\gg n^{3-o(1)}$, see [166].) Bradać, Gishboliner, and Sudakov [BGS23] have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \\ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.\n\nThis problem is #32 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#567: [EFRS93][Er95,p.177]" }, { "number": 568, "url": "https://www.erdosproblems.com/568", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,\\[R(G,H)\\ll m?\\]", "commentary": "In other words, is $G$ Ramsey size linear?\n\nThis problem is #33 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#568: [EFRS93]" }, { "number": 569, "url": "https://www.erdosproblems.com/569", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that\\[R(C_{2k+1},H)\\leq c_k m\\]for any graph $H$ on $m$ edges without isolated vertices?", "commentary": "This problem is #34 in Ramsey Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#569: [EFRS93]" }, { "number": 571, "url": "https://www.erdosproblems.com/571", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Show that for any rational $\\alpha \\in [1,2)$ there exists a bipartite graph $G$ such that\\[\\mathrm{ex}(n;G)\\asymp n^{\\alpha}.\\]", "commentary": "A problem of Erdős and Simonovits. In [Er78] Erdős wrote 'I am not entirely sure that a trivial counterexample can not be found'. \n\nBukh and Conlon [BuCo18] proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.\n\nA rational $\\alpha\\in [1,2)$ for which this holds is known as a Turán exponent. Known Turán exponents are:\n $\\frac{3}{2}-\\frac{1}{2s}$ for $s\\geq 2$ (Conlon, Janzer, and Lee [CJL21]).\n $\\frac{4}{3}-\\frac{1}{3s}$ and $\\frac{5}{4}-\\frac{1}{4s}$ for $s\\geq 2$ (Jiang and Qiu [JiQi20]).\n $2-\\frac{a}{b}$ for $\\lfloor b/a\\rfloor^3 \\leq a\\leq \\frac{b}{\\lfloor b/a\\rfloor+1}+1$ (Jiang, Jiang, and Ma [JJM20]).\n $2-\\frac{a}{b}$ with $b>a\\geq 1$ and $b\\equiv \\pm 1\\pmod{a}$ (Kang, Kim, and Liu [KKL21]).\n $1+a/b$ with $b>a^2$ (Jiang and Qiu [JiQi23]),\n $2-\\frac{2}{2b+1}$ for $b\\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan [JMY22]).\n $2-a/b$ with $b\\geq (a-1)^2$ (Conlon and Janzer [CoJa22]).\nSee also [713].\n\nThis problem is #45 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#571: [Er74c,p.78][Er75][Er78,p.30][Er81][ErSi84][Er91]" }, { "number": 572, "url": "https://www.erdosproblems.com/572", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Show that for $k\\geq 3$\\[\\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}.\\]", "commentary": "It is easy to see that $\\mathrm{ex}(n;C_{2k+1})=\\lfloor n^2/4\\rfloor$ for any $k\\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erdős and Klein [Er38] proved $\\mathrm{ex}(n;C_4)\\asymp n^{3/2}$. \n\nErdős [Er64c] and Bondy and Simonovits [BoSi74] showed that\\[\\mathrm{ex}(n;C_{2k})\\ll kn^{1+\\frac{1}{k}}.\\]Benson [Be66] has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar [LUW95] have shown that, for arbitrary $k\\geq 3$,\\[\\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{2}{3k-3+\\nu}},\\]where $\\nu=0$ if $k$ is odd and $\\nu=1$ if $k$ is even. See [LUW99] for further history and references.\n\nSee also [765].\n\nThis problem is #46 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#572: [Er64c][Er71,p.103][Er74c,p.78]" }, { "number": 573, "url": "https://www.erdosproblems.com/573", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [ "A006856" ], "formalized": "no", "statement": "Is it true that\\[\\mathrm{ex}(n;\\{C_3,C_4\\})\\sim (n/2)^{3/2}?\\]", "commentary": "A problem of Erdős and Simonovits, who proved that\\[\\mathrm{ex}(n;\\{C_4,C_5\\})=(n/2)^{3/2}+O(n).\\]Kövári, Sós, and Turán [KST54] proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\\sim (n/2)^{3/2}$. This problem is therefore asking whether the threshold is the same if we just forbid odd cycles of length $3$.\n\nSee also [574] for the general case, and [765] for $\\mathrm{ex}(n;C_4)$.\n\nThis problem is #48 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#573: [Er71,p.103][Er75][ErSi82][Er93,p.336]" }, { "number": 575, "url": "https://www.erdosproblems.com/575", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$. Is it true that, for every $\\mathcal{F}$, if there is a bipartite graph in $\\mathcal{F}$ then there exists some bipartite $G\\in\\mathcal{F}$ such that\\[\\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})?\\]", "commentary": "A problem of Erdős and Simonovits.\n\nSee also [180].\n\nThis problem is #51 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#575: [ErSi82]" }, { "number": 576, "url": "https://www.erdosproblems.com/576", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of\\[\\mathrm{ex}(n;Q_k).\\]", "commentary": "Erdős and Simonovits [ErSi70] proved that\\[(\\tfrac{1}{2}+o(1))n^{3/2}\\leq \\mathrm{ex}(n;Q_3) \\ll n^{8/5}.\\](In [ErSi70] they mention that Erdős had originally conjectured that $ \\mathrm{ex}(n;Q_3)\\gg n^{5/3}$.) Erdős and Simonovits also proved that, if $G$ is the graph $Q_3$ with a missing edge, then $\\mathrm{ex}(n;G)\\asymp n^{3/2}$.\n\nIn [Er74c], [Er81], and [Er93] Erdős asked whether it is $\\mathrm{ex}(n;Q_3)\\asymp n^{8/5}$.\n\nA theorem of Sudakov and Tomon [SuTo22] implies\\[\\mathrm{ex}(n;Q_k)=o(n^{2-\\frac{1}{k}}).\\]Janzer and Sudakov [JaSu22] have improved this to\\[\\mathrm{ex}(n;Q_k)\\ll_k n^{2-\\frac{1}{k-1}+\\frac{1}{(k-1)2^{k-1}}}.\\]See also [1035].\n\nThis problem is #52 in Extremal Graph Theory in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#576: [Er64c][ErSi70,p.378][Er74c,p.78][Er75][Er81][Er93,p.334]" }, { "number": 579, "url": "https://www.erdosproblems.com/579", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\\delta n^2$ edges then $G$ contains an independent set of size $\\gg_\\delta n$.", "commentary": "A problem of Erdős, Hajnal, Sós, and Szemerédi, who could prove this is true for $\\delta>1/8$.\n\nSee also [533] and the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#579: [EHSS83][Er90][Er91][Er93,p.340]" }, { "number": 580, "url": "https://www.erdosproblems.com/580", "status": "decidable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $n/2$ vertices?", "commentary": "A conjecture of Erdős, Füredi, Loebl, and Sós. Ajtai, Komlós, and Szemerédi [AKS95] proved an asymptotic version, where at least $(1+\\epsilon)n/2$ vertices have degree at least $(1+\\epsilon)n/2$ (and $n$ is sufficiently large depending on $\\epsilon$).\n\nZhao [Zh11] has proved this conjecture holds for all sufficiently large $n$. \n\nKomlós and Sós conjectured the generalisation that if at least $n/2$ vertices have degree at least $k$ then $G$ contains any tree with $k$ vertices.\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 October 2025. View history", "references": "#580: [EFLS95]" }, { "number": 583, "url": "https://www.erdosproblems.com/583", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Every connected graph on $n$ vertices can be partitioned into at most $\\lceil n/2\\rceil$ edge-disjoint paths.", "commentary": "A problem of Erdős and Gallai. If we drop the edge-disjoint condition then this conjecture was proved by Fan [Fa02].\n\nThere are many partial results towards this conjecture. We list some highlights below; $G$ is an arbitrary connected graph on $n$ vertices.\n Lovász [Lo68] proved that $G$ can be partitioned into at most $\\lfloor n/2\\rfloor$ edge-disjoint paths and cycles, and hence into at most $n-1$ paths. This also implies the original conjecture if $G$ has at most one vertex of even degree. \n Chung [Ch78] proved that $G$ can be partitioned into at most $\\lceil n/2\\rceil$ edge-disjoint trees.\n Pyber [Py96] proved that $G$ can be covered by at most $n/2+O(n^{3/4})$ paths (that may not be edge-disjoint).\n Pyber [Py96] also proved the original conjecture if the subgraph induced by vertices of even degree is a forest.\n Bonamy and Perrett [BoPe19] proved the conjecture if $G$ has maximum degree $\\leq 5$.\n Blanché, Bonamy, and Bonichon [BBB21] proved this conjecture if $G$ is planar.\n Anto and Basavaraju [AnBa23] proved the conjecture if $G$ is $2$-degenerate (every subgraph has a vertex of degree $\\leq 2$).\n Chu, Fan, and Zhou [CFZ26] proved the conjecture if the subgraph induced by vertices of even degree is $K_m$ for some $m\\leq 15$ and $n$ is odd.\n Dean and Kouider [DeKo00] prove that $\\lceil \\frac{2}{3}n\\rceil$ edge-disjoint paths always suffice, even for disconnected graphs (for which this bound is optimal). The same bound was independently obtained by Yan in their 1998 PhD thesis.\nHajós [Lo68] has conjectured that if $G$ has all degrees even then $G$ can be partitioned into at most $\\lfloor n/2\\rfloor$ edge-disjoint cycles.\n\nSee also [184] for an analogous problem decomposing into edges and cycles and [1017] for decomposing into complete graphs. See also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#583: [Er71,p.101]" }, { "number": 584, "url": "https://www.erdosproblems.com/584", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph with $n$ vertices and $\\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\\subseteq G$ such that\n$H_1$ has $\\gg \\delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and\n$H_2$ has $\\gg \\delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.", "commentary": "A problem of Erdős, Duke, and Rödl. Duke and Erdős [DuEr82] proved the first if $n$ is sufficiently large depending on $\\delta$. The real challenge is to prove this when $\\delta=n^{-c}$ for some $c>0$. Duke, Erdős, and Rödl [DER84] proved the first statement with a $\\delta^5$ in place of a $\\delta^3$.\n\nFox and Sudakov [FoSu08b] have proved the second statement when $\\delta >n^{-1/5}$. \n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#584: [DuEr82][DER84]" }, { "number": 585, "url": "https://www.erdosproblems.com/585", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "no", "statement": "What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?", "commentary": "Pyber, Rödl, and Szemerédi [PRS95] constructed such a graph with $\\gg n\\log\\log n$ edges.\n\nChakraborti, Janzer, Methuku, and Montgomery [CJMM24] have shown that such a graph can have at most $n(\\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#585: [Er76b]" }, { "number": 588, "url": "https://www.erdosproblems.com/588", "status": "open", "prize": "$100", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $f_k(n)$ be minimal such that if $n$ points in $\\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that\\[f_k(n)=o(n^2)\\]for $k\\geq 4$?", "commentary": "A generalisation of [101] (which asks about $k=4$). \n\nThe restriction to $k\\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84] for constructions which show that $f_3(n)\\geq(1/6+o(1))n^2$.)\n\nFor $k\\geq 4$, Kárteszi [Ka63] proved\\[f_k(n)\\gg_k n\\log n\\](resolving a conjecture of Erdős that $f_k(n)/n\\to \\infty$). Grünbaum [Gr76] proved\\[f_k(n) \\gg_k n^{1+\\frac{1}{k-2}}.\\]Erdős speculated this may be the correct order of magnitude, but Solymosi and Stojaković [SoSt13] give a construction which shows\\[f_k(n)\\gg_k n^{2-O_k(1/\\sqrt{\\log n})}\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#588: [Er84]" }, { "number": 589, "url": "https://www.erdosproblems.com/589", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $g(n)$ be maximal such that in any set of $n$ points in $\\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.", "commentary": "The trivial greedy algorithm gives $g(n)\\gg n^{1/2}$. A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\\leq l\\aleph_0$.", "commentary": "Similar problems were investigated by Erdős, Galvin, and Hajnal [EGH75]. Erdős claims that for graphs the problem is completely solved: a graph of chromatic number $\\geq \\aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#593: [Er95d]" }, { "number": 595, "url": "https://www.erdosproblems.com/595", "status": "open", "prize": "$250", "tags": [ "graph theory", "set theory" ], "oeis": [], "formalized": "no", "statement": "Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?", "commentary": "A problem of Erdős and Hajnal. Folkman [Fo70] and Nešetřil and Rödl [NeRo75] have proved that for every $n\\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.\n\nSee also [582] and [596].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#595: [Er87]" }, { "number": 596, "url": "https://www.erdosproblems.com/596", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory", "set theory" ], "oeis": [], "formalized": "no", "statement": "For which graphs $G_1,G_2$ is it true that\n for every $n\\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet\n for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.", "commentary": "Erdős and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Nešetřil and Rödl established the first property and Erdős and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees). \n\nWhether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#596: [Er87]" }, { "number": 597, "url": "https://www.erdosproblems.com/597", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory", "set theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph on at most $\\aleph_1$ vertices which contains no $K_4$ and no $K_{\\aleph_0,\\aleph_0}$ (the complete bipartite graph with $\\aleph_0$ vertices in each class). Is it true that\\[\\omega_1^2 \\to (\\omega_1\\omega, G)^2?\\]What about finite $G$?", "commentary": "Erdős and Hajnal proved that $\\omega_1^2 \\to (\\omega_1\\omega,3)^2$. Erdős originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\\omega_1^2 \\not\\to (\\omega_1\\omega, K_{\\aleph_0,\\aleph_0})^2$. \n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#597: [Er87][Va99,7.84]" }, { "number": 598, "url": "https://www.erdosproblems.com/598", "status": "open", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colours so that every $X\\subseteq m$ with $\\lvert X\\rvert=\\kappa$ contains subsets of all possible colours?", "commentary": "View the LaTeX source\n\n\n\n\n \n View history", "references": "#598: [Er87]" }, { "number": 600, "url": "https://www.erdosproblems.com/600", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\\geq 2$. Is it true that\\[e(n,r+1)-e(n,r)\\to \\infty\\]as $n\\to \\infty$? Is it true that\\[\\frac{e(n,r+1)}{e(n,r)}\\to 1\\]as $n\\to \\infty$?", "commentary": "Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$.\n\nSee also [80].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#600: [Er87]" }, { "number": 601, "url": "https://www.erdosproblems.com/601", "status": "open", "prize": "$500", "tags": [ "graph theory", "set theory" ], "oeis": [], "formalized": "no", "statement": "For which limit ordinals $\\alpha$ is it true that if $G$ is a graph with vertex set $\\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\\alpha$?", "commentary": "A problem of Erdős, Hajnal, and Milner [EHM70], who proved this is true for $\\alpha < \\omega_1^{\\omega+2}$. \n\nIn [Er82e] Erdős offers \\$250 for showing what happens when $\\alpha=\\omega_1^{\\omega+2}$ and \\$500 for settling the general case.\n\nLarson [La90] proved this is true for all $\\alpha<2^{\\aleph_0}$ assuming Martin's axiom.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#601: [EHM70][Er81][Er82e][Er87]" }, { "number": 602, "url": "https://www.erdosproblems.com/602", "status": "open", "prize": "no", "tags": [ "combinatorics", "set theory" ], "oeis": [], "formalized": "no", "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\\neq 1$. Is there a $2$-colouring of $\\cup A_i$ such that no $A_i$ is monochromatic?", "commentary": "A problem of Komjáth. The existence of such a $2$-colouring is sometimes known as Property B.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#602: [Er87]" }, { "number": 603, "url": "https://www.erdosproblems.com/603", "status": "open", "prize": "no", "tags": [ "combinatorics", "set theory" ], "oeis": [], "formalized": "no", "statement": "Let $(A_i)$ be a family of countably infinite sets such that $\\lvert A_i\\cap A_j\\rvert \\neq 2$ for all $i\\neq j$. Find the smallest cardinal $C$ such that $\\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.", "commentary": "A problem of Komjáth. If instead we have $\\lvert A_i\\cap A_j\\rvert \\neq 1$ then Komjáth showed that this is possible with at most $\\aleph_0$ colours.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#603: [Er87]" }, { "number": 604, "url": "https://www.erdosproblems.com/604", "status": "open", "prize": "$500", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Given $n$ distinct points $A\\subset\\mathbb{R}^2$ must there be a point $x\\in A$ such that\\[\\#\\{ d(x,y) : y \\in A\\} \\gg n^{1-o(1)}?\\]Or even $\\gg n/\\sqrt{\\log n}$?", "commentary": "The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\\sqrt{\\log n}$ would be best possible.\n\nIt may be true that there are $\\gg n$ many such points (Hunter has noted in the comments that this trivially follows from the existence of a single point), or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in [Er75f] Erdős conjectured\\[\\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}},\\]where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\n\nIn [Er97e] Erdős offers \\$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\\gg n$ many such points.\n\nIn [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.\n\nThe best known bound is\\[\\gg n^{c-o(1)},\\]due to Katz and Tardos [KaTa04], where\\[c=\\frac{48-14e}{55-16e}=0.864137\\cdots.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#604: [Er57][Er61][Er75f,p.99][Er83c][Er85][Er87b,p.169][Er90][Er95][Er97b][Er97c][Er97e][Er97f]" }, { "number": 609, "url": "https://www.erdosproblems.com/609", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.", "commentary": "A problem of Erdős and Graham. The edges of $K_{2^n}$ can be $n$-coloured to avoid odd cycles of any length. It can be shown that $C_5$ and $C_7$ can be avoided for large $n$.\n\nChung [Ch97] asked whether $f(n)\\to \\infty$ as $n\\to \\infty$. Day and Johnson [DaJo17] proved this is true, and that\\[f(n)\\geq 2^{c\\sqrt{\\log n}}\\]for some constant $c>0$. The trivial upper bound is $2^n$.\n\nGirão and Hunter [GiHu24] have proved that\\[f(n) \\ll \\frac{2^n}{n^{1-o(1)}}.\\]Janzer and Yip [JaYi25] have improved this to\\[f(n) \\ll n^{3/2}2^{n/2}.\\]See also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#609: [ErGr75]" }, { "number": 610, "url": "https://www.erdosproblems.com/610", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).Estimate $\\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then\\[\\tau(G) \\leq n-\\omega(n)\\sqrt{n}\\]for some $\\omega(n)\\to \\infty$, or even\\[\\tau(G) \\leq n-c\\sqrt{n\\log n}\\]for some absolute constant $c>0$?", "commentary": "A problem of Erdős, Gallai, and Tuza [EGT92], who proved that\\[\\tau(G) \\leq n-\\sqrt{2n}+O(1).\\]This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\\sqrt{n\\log n})$, which follows from the lower bound for $R(3,k)$ by Kim [Ki95] (see [165]).\n\nIndeed, Erdős, Gallai, and Tuza speculate that if $f(n)$ is the largest $k$ such that every triangle-free graph on $n$ vertices contains an independent set on $f(n)$ vertices, then $\\tau(G)\\leq n-f(n)$.\n\nA positive answer to this problem would follow from a positive answer to [151] (since Ajtai, Komlós, and Szemerédi [AKS80] have proved that the $H(n)$ defined there satisfies $H(n)\\gg \\sqrt{n\\log n}$).\n\nSee also [151], [611], this entry and and this entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#610: [EGT92][Er94][Er99]" }, { "number": 611, "url": "https://www.erdosproblems.com/611", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\\tau(G)=o_c(n)$?Similarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\\tau(G)<(1-c)n$.", "commentary": "A problem of Erdős, Gallai, and Tuza [EGT92], who proved for the latter question that $k_c(n) \\geq n^{c'/\\log\\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\\tau(G) \\leq n-(kn)^{1/2}$. Bollobás and Erdős proved that if every maximal clique has at least $n+3-2\\sqrt{n}$ vertices then $\\tau(G)=1$ (and this threshold is best possible).\n\nSee also [610] and the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#611: [EGT92][Er94][Er99]" }, { "number": 612, "url": "https://www.erdosproblems.com/612", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then\\[D\\leq \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1),\\]and if $G$ contains no $K_{2r+1}$ and $3r-1 \\mid d$ then\\[D\\leq \\frac{3r-1}{r}\\frac{n}{d}+O(1).\\]", "commentary": "A problem of Erdős, Pach, Pollack, and Tuza [EPPT89], who gave constructions showing that the above bounds would be sharp, and proved the case $2r+1=3$. It is known (see [EPPT89] for example) that any connected graph on $n$ vertices with minimum degree $d$ has diameter\\[D\\leq 3\\frac{n}{d+1}+O(1).\\]This was disproven for the case of $K_{2r}$-free graphs with $r\\geq 2$ by Czabarka, Singgih, and Székely [CSS21], who constructed arbitrarily large connected graphs on $n$ vertices which contain no $K_{2r}$ and have minimum degree $d$, and diameter\\[\\frac{6r-5}{(2r-1)d+2r-3}n+O(1),\\]which contradicts the above conjecture for each fixed $r$ as $d\\to \\infty$.\n\nThey suggest the amended conjecture, which no longer divides into two cases, that if $G$ is a connected graph on $n$ vertices with minimum degree $d$ which contains no $K_{k+1}$ then the diameter of $G$ is at most\\[(3-\\tfrac{2}{k})\\frac{n}{d}+O(1).\\]This bound is known under the weaker assumption that $G$ is $k$-colourable when $k=3$ and $k=4$, shown by Czabarka, Dankelmann, and Székely [CDS09] and Czabarka, Smith, and Székely [CSS23].\n\nCambie and Jooken [CaJo25] have given an example that shows the diameter for $K_4$-free graphs with minimum degree $16$ is at least $\\frac{31}{216}n+O(1)$, giving another counterexample to the original conjecture.\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#612: [EPPT89]" }, { "number": 614, "url": "https://www.erdosproblems.com/614", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.", "commentary": "See also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#614: [FRS97]" }, { "number": 616, "url": "https://www.erdosproblems.com/616", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 3$. For an $r$-uniform hypergraph $G$ let $\\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.Determine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\\tau(G')\\leq 1$, we have $\\tau(G)\\leq t$.", "commentary": "Erdős, Hajnal, and Tuza [EHT91] proved that this $t$ satisfies\\[\\frac{3}{16}r+\\frac{7}{8}\\leq t \\leq \\frac{1}{5}r.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#616: [Er99]" }, { "number": 617, "url": "https://www.erdosproblems.com/617", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "yes", "statement": "Let $r\\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.", "commentary": "In other words, there is no balanced colouring. A conjecture of Erdős and Gyárfás [ErGy99], who proved it for $r=3$ and $r=4$ (and observed it is false for $r=2$), and showed this property fails for infinitely many $r$ if we replace $r^2+1$ by $r^2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#617: [ErGy99][Er99]" }, { "number": 619, "url": "https://www.erdosproblems.com/619", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).Is it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?", "commentary": "A problem of Erdős, Gyárfás, and Ruszinkó [EGR98] who proved that $h_3(G)\\leq n$ and $h_5(G) \\leq \\frac{n-1}{2}$ and there exist connected graphs $G$ on $n$ vertices with $h_3(G)\\geq n-c$ for some constant $c>0$.\n\nIf we omit the condition that the graph must remain triangle-free then Alon, Gyárfás, and Ruszinkó [AGR00] have proved that adding $n/2$ edges always suffices to obtain diameter at most $4$.\n\nSee also [134] and [618].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#619: [EGR98][Er99]" }, { "number": 620, "url": "https://www.erdosproblems.com/620", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?", "commentary": "This was first asked by Erdős and Rogers [ErRo62], and is generally known as the Erdős-Rogers problem. Let $f(n)$ be such that every such graph contains a triangle-free subgraph with at least $f(n)$ vertices.\n\nIt is now known that $f(n)=n^{1/2+o(1)}$. Bollobás and Hind [BoHi91] proved\\[n^{1/2} \\ll f(n) \\ll n^{7/10+o(1)}.\\]Krivelevich [Kr94] improved this to\\[n^{1/2}(\\log\\log n)^{1/2} \\ll f(n) \\ll n^{2/3}(\\log n)^{1/3}.\\]Wolfovitz [Wo13] proved\\[f(n) \\ll n^{1/2}(\\log n)^{120}.\\]The best bounds currently known are\\[n^{1/2}\\frac{(\\log n)^{1/2}}{\\log\\log n}\\ll f(n) \\ll n^{1/2}\\log n.\\]The lower bound follows from results of Shearer [Sh95], and the upper bound was proved by Mubayi and Verstraete [MuVe24].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#620: [ErRo62][EGT92][Er99]" }, { "number": 623, "url": "https://www.erdosproblems.com/623", "status": "open", "prize": "no", "tags": [ "set theory" ], "oeis": [], "formalized": "yes", "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$ that is independent - that is, for all finite $B\\subset Y$ we have $f(B)\\not\\in Y$?", "commentary": "A problem of Erdős and Hajnal [ErHa58], who proved that if $\\lvert X\\rvert <\\aleph_\\omega$ then the answer is no. Erdős suggests in [Er99] that this problem is 'perhaps undecidable'. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#623: [ErHa58][Er99]" }, { "number": 624, "url": "https://www.erdosproblems.com/624", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$ we have\\[\\{ f(A) : A\\subseteq Y\\}=X.\\]Prove that\\[H(n)-\\log_2 n \\to \\infty.\\]", "commentary": "A problem of Erdős and Hajnal [ErHa68] who proved that\\[\\log_2 n \\leq H(n) < \\log_2n +(3+o(1))\\log_2\\log_2n.\\]Erdős said that even the weaker statement that for $n=2^k$ we have $H(n)\\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\\frac{n-1}{2}$ subsets $A_i$ of size $2$ such that $f(A_i)$ is the same. Any set $Y$ of size $k$ containing at least $k/2$ of them can have at most\\[2^k-\\lfloor k/2\\rfloor+1< 2^k=n\\]distinct elements in the union of the images of $f(A)$ for $A\\subseteq Y$. \n\nFor this weaker statement, Erdős and Gyárfás conjectured the stronger form that if $\\lvert X\\rvert=2^k$ then, for any $f:\\{A : A\\subseteq X\\}\\to X$, there must exist some $Y\\subset X$ of size $k$ such that\\[\\#\\{ f(A) : A\\subseteq Y\\}< 2^k-k^C\\]for every $C$ (with $k$ sufficiently large depending on $C$). This was proved by Alon (personal communication), who proved the stronger version that there exists some absolute constant $c>0$ such that, if $k$ is large enough, there must exist some $Y\\subset X$ of size $k$ such that\\[\\#\\{ f(A) : A\\subseteq Y\\}<(1-c)2^k.\\]Alon also proved that, provided $k$ is large enough, if $\\lvert X\\rvert=2^k$ there exists some $f:\\{A: A\\subseteq X\\}\\to X$ such that, if $Y\\subset X$ with $\\lvert Y\\rvert=k$, then\\[\\#\\{ f(A) : A\\subseteq Y\\}>\\tfrac{1}{4}2^k.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 October 2025. View history", "references": "#624: [ErHa68][Er99]" }, { "number": 625, "url": "https://www.erdosproblems.com/625", "status": "open", "prize": "$1000", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\\chi(G)$ denote the chromatic number.If $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely\\[\\chi(G) - \\zeta(G) \\to \\infty\\]as $n\\to \\infty$?", "commentary": "A problem of Erdős and Gimbel (see also [Gi16]). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erdős offered \\$100 for a proof that this is true, and \\$1000 for a proof that this is false (although later told Gimbel that \\$1000 was perhaps too much).\n\nIt is known that almost surely\\[\\frac{n}{2\\log_2n}\\leq \\zeta(G)\\leq \\chi(G)\\leq (1+o(1))\\frac{n}{2\\log_2n}.\\](The final upper bound is due to Bollobás [Bo88]. The first inequality follows from the fact that almost surely $G$ has clique number and independence number $< 2\\log_2n$.)\n\nHeckel [He24] and, independently, Steiner [St24b] have shown that it is not the case that $\\chi(G)-\\zeta(G)$ is bounded with high probability, and in fact if $\\chi(G)-\\zeta(G) \\leq f(n)$ with high probability then $f(n)\\geq n^{1/2-o(1)}$ along an infinite sequence of $n$. Heckel conjectures that, with high probability,\\[\\chi(G)-\\zeta(G) \\asymp \\frac{n}{(\\log n)^3}.\\]Heckel [He24c] further proved that, for any $\\epsilon>0$, we have\\[\\chi(G) -\\zeta(G) \\geq n^{1-\\epsilon}\\]for roughly $95\\%$ of all $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#625: [ErGi93]" }, { "number": 626, "url": "https://www.erdosproblems.com/626", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number", "cycles" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\\leq m$). Does\\[\\lim_{n\\to \\infty}\\frac{g_k(n)}{\\log n}\\]exist?Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does\\[\\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\]exist, and what is its value?", "commentary": "It is known that\\[\\frac{1}{4\\log k}\\log n\\leq g_k(n) \\leq \\frac{2}{\\log(k-2)}\\log n+1,\\]the lower bound due to Kostochka [Ko88] and the upper bound to Erdős [Er59b]. \n\nErdős [Er59b] proved that\\[\\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\gg \\frac{1}{m}\\]and, for odd $m$,\\[\\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\leq \\frac{2}{m+1},\\]and conjectured this is sharp. He had no good guess for the value of the limit for even $m$, other that it should lie in $[\\frac{2}{m+2},\\frac{2}{m}]$, but could not prove this even for $m=4$.\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#626: [Er59b][Er62b][Er69b]" }, { "number": 627, "url": "https://www.erdosproblems.com/627", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $\\omega(G)$ denote the clique number of $G$ and $\\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\\chi(G)/\\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does\\[\\lim_{n\\to\\infty}\\frac{f(n)}{n/(\\log_2n)^2}\\]exist?", "commentary": "Tutte and Zykov [Zy52] independently proved that for every $k$ there is a graph with $\\omega(G)=2$ and $\\chi(G)=k$. Erdős [Er61d] proved that for every $n$ there is a graph on $n$ vertices with $\\omega(G)=2$ and $\\chi(G)\\gg n^{1/2}/\\log n$, whence $f(n) \\gg n^{1/2}/\\log n$.\n\nErdős [Er67c] proved that\\[f(n) \\asymp \\frac{n}{(\\log_2n)^2}\\]and that the limit in question, if it exists, must be in $[1/4,4]$. (The upper bound he states as $1$, but [AFM25] note the method actually gives $4$.)\n\nAraujo, Filipe, and Miyazaki [AFM25] prove that if $\\lim\\frac{\\log R(k)}{k}$ exists and is equal to $C$ (see [77]), and also $R(s,t)\\leq R(k)$ for all $st\\leq k^2$, then the limit in this question also exists and is equal to $C^2$. Using this connection they improve the upper bound from $4$ to $\\approx 3.7$.\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#627: [Er61d][Er67c][Er69b]" }, { "number": 628, "url": "https://www.erdosproblems.com/628", "status": "falsifiable", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\\geq a$ and $\\geq b$ respectively?", "commentary": "This property is sometimes called being $(a,b)$-splittable. A question of Erdős and Lovász (often called the Erdős-Lovász Tihany conjecture). Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles).\n\nBalogh, Kostochka, Prince, and Stiebitz [BKPS09] have proved the full conjecture for quasi-line graphs and graphs with independence number $2$.\n\nFor more partial results in this direction see the comprehensive survey of this problem by Song [So22].\n\nSee also the entry in the graphs problem collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#628: [Er68b]" }, { "number": 629, "url": "https://www.erdosproblems.com/629", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.Determine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\\chi_L(G)>k$.", "commentary": "A problem of Erdős, Rubin, and Taylor [ERT80], who proved that\\[2^{k-1}0$: take $A$ to be the union of all odd numbers together with numbers of the shape $2^k$ with $k$ odd.\n\nThe second question has been resolved in the affirmative by ChatGPT-5.2 (prompted by Leeham); Tao has since observed that a positive answer also follows fairly quickly from an inequality due to Elliott [El79].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#635: [Gu83][Ru99]" }, { "number": 638, "url": "https://www.erdosproblems.com/638", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.Is it true that for every infinite cardinal $\\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\\aleph$ many colours then there is a monochromatic triangle.", "commentary": "Erdős writes 'if the answer is affirmative many extensions and generalisations will be possible'.\n\nBarreto notes in the comments that (as is often the case with problems of this kind) presumably $S$ is intended to also be closed under taking subgraphs, or else a sparse family of complete graphs gives a trivial counterexample.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#638: [Er97d]" }, { "number": 640, "url": "https://www.erdosproblems.com/640", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 3$. Does there exist some $f(k)$ such that if a graph $G$ has chromatic number $\\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\\geq k$?", "commentary": "A problem of Erdős and Hajnal. \n\nThis is trivial when $k=3$, since any non-bipartite graph contains an odd cycle, and all odd cycles have chromatic number $3$.\n\nSteiner has observed in the comments that this is equivalent to the problem in which we replace 'odd cycle' with 'path'.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#640: [Er97d,p.84]" }, { "number": 642, "url": "https://www.erdosproblems.com/642", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than chords. Is it true that $f(n)\\ll n$?", "commentary": "A problem of Hamburger and Szegedy. A chord is an edge between two vertices of the cycle which are not consecutive in the cycle.\n\nChen, Erdős, and Staton [CES96] proved $f(n) \\ll n^{3/2}$. Draganić, Methuku, Munhá Correia, and Sudakov [DMMS24] have improved this to\\[f(n) \\ll n(\\log n)^8.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 January 2026. View history", "references": "#642: [CES96][Er97d,p.84]" }, { "number": 643, "url": "https://www.erdosproblems.com/643", "status": "open", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that\\[A\\cup B= C\\cup D\\]and\\[A\\cap B=C\\cap D=\\emptyset.\\]Estimate $f(n;t)$ - in particular, is it true that for $t\\geq 3$\\[f(n;t)=(1+o(1))\\binom{n}{t-1}?\\]", "commentary": "For $t=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1/2+o(1))n^{3/2}$. \n\nFüredi [Fu84] proved that $f(n;3) \\ll n^2$ and $f(n;3) > \\binom{n}{2}$ for infinitely many $n$. Pikhurko and Verstraëte [PiVe09] have proved $f(n;3)\\leq \\frac{13}{9}\\binom{n}{2}$ for all $n$.\n\nMore generally, Füredi [Fu84] proved that\\[\\binom{n-1}{t-1}+\\left\\lfloor\\frac{n-1}{t}\\right\\rfloor\\leq f(n;t) < \\frac{7}{2}\\binom{n}{t-1},\\]and conjectured the lower bound is sharp for $t\\geq 4$. Pikhurko and Verstraëte [PiVe09] have proved that\\[1 \\leq \\limsup_{n\\to \\infty} \\frac{f(n;t)}{\\binom{n}{t-1}}\\leq \\min\\left(\\frac{7}{4},1+\\frac{2}{\\sqrt{t}}\\right)\\]for all $t\\geq 3$.\n\nFüredi [Fu84] proved that $f(n;3)/\\binom{n}{2}$ converges as $n\\to \\infty$, but the existence of the limit for $t\\geq 4$ is unknown.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 October 2025. View history", "references": "#643: [Er77b][Er97d]" }, { "number": 644, "url": "https://www.erdosproblems.com/644", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $f(k,r)$ be minimal such that if $A_1,A_2,\\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\\{x,y\\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that\\[f(k,7)=(1+o(1))\\frac{3}{4}k?\\]Is it true that for any $r\\geq 3$ there exists some constant $c_r$ such that\\[f(k,r)=(1+o(1))c_rk?\\]", "commentary": "A problem of Erdős, Fon-Der-Flaass, Kostochka, and Tuza [EFKT92], who proved that $f(k,3)=2k$ and $f(k,4)=\\lfloor 3k/2\\rfloor$ and $f(k,5)=\\lfloor 5k/4\\rfloor$, and further that $f(k,6)=k$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#644: [EFKT92][Er97d]" }, { "number": 647, "url": "https://www.erdosproblems.com/647", "status": "verifiable", "prize": "£25", "tags": [ "number theory" ], "oeis": [ "A062249", "A087280" ], "formalized": "yes", "statement": "Let $\\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\\[\\max_{m24$. He wrote 'I am being rather stingy but we old people are stingy.' (This has been converted to \\$44 using approximate 1992 exchange rates.)\n\nTao has observed in the comments that, since $\\tau(m)$ is similar to $2^{\\omega(m)}$, this problem is similar to (but slightly weaker than) the first part of [679], but much stronger than [413] or [248].\n\nSee also [413].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#647: [Er79][Er79d][Er80,p.107][Er92e][Er95c]" }, { "number": 653, "url": "https://www.erdosproblems.com/653", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\\neq i\\}$, where the points are ordered such that\\[R(x_1)\\leq \\cdots \\leq R(x_n).\\]Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \\geq (1-o(1))n$?", "commentary": "Erdős and Fishburn proved $g(n)>\\frac{3}{8}n$ and Csizmadia proved $g(n)>\\frac{7}{10}n$. Both groups proved $g(n) < n-cn^{2/3}$ for some constant $c>0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#653: [Er97e]" }, { "number": 654, "url": "https://www.erdosproblems.com/654", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be such that, given any $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ with no four points on a circle, there exists some $x_i$ with at least $f(n)$ many distinct distances to other $x_j$. Estimate $f(n)$ - in particular, is it true that\\[f(n)>(1-o(1))n?\\]Or at least\\[f(n) > (1/3+c)n\\]for some $c>0$, for all large $n$?", "commentary": "It is trivial that $f(n) \\geq (n-1)/3$. In [Er97e] Erdős asks whether $f(n)>(1-o(1))n$, but says this is 'perhaps too optimistic but I am sure that [the trivial bound] can be significantly improved'.\n\nIn [Er87b] and [ErPa90] Erdős and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker question whether there is a lower bound of the shape $(\\tfrac{1}{3}+c)n$ for some constant $c>0$.\n\nThey suggest the lower bound $(1-o(1))n$ is true under the assumption that any circle around a point $x_i$ contains at most $2$ other $x_j$.\n\nThe strongest form of this conjecture was disproved by Aletheia [Fe26], which constructed a set of $n$ points in $\\mathbb{R}^2$, no four on a circle, such that there are at most $\\frac{3}{4}n$ distinct distances from any single point (although this construction has all the points on the union of two lines, so does not help with the stronger version with the points in general position).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#654: [Er87b,p.168][ErPa90,p.267][Er97e,p.530]" }, { "number": 655, "url": "https://www.erdosproblems.com/655", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least\\[(1+c)\\frac{n}{2}\\]distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?", "commentary": "A problem of Erdős and Pach. It is easy to see that this assumption implies that there are at least $\\frac{n-1}{2}$ distinct distances determined by every point.\n\nZach Hunter has observed that taking $n$ points equally spaced on a circle disproves this conjecture. In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#655: [Er97e]" }, { "number": 657, "url": "https://www.erdosproblems.com/657", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Is it true that if $A\\subset \\mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\\to \\infty$?", "commentary": "In [Er73] Erdős attributes this problem (more generally in $\\mathbb{R}^k$) to himself and Davies. In [Er97e] he does not mention Davis, but says this problem was investigated by himself, Füredi, Ruzsa, and Pach.\n\nIn [Er73] Erdős says it is not even known in $\\mathbb{R}$ whether $f(n)\\to \\infty$. Sarosh Adenwalla has observed that this is equivalent to minimising the number of distinct differences in a set $A\\subset \\mathbb{R}$ of size $n$ without three-term arithmetic progressions. Dumitrescu [Du08] proved that, in these terms,\\[(\\log n)^c \\leq f(n) \\leq 2^{O(\\sqrt{\\log n})}\\]for some constant $c>0$.\n\nHunter observed in the comments that a result of Ruzsa coupled with standard tools of additive combinatorics (with details given by Alfaiz and Tang) allow recent progress on the size of subsets without three-term arithmetic progression (see [BlSi23] which improves slightly on the bounds due to Kelley and Meka [KeMe23]) yield\\[2^{c(\\log n)^{1/9}}\\leq f(n)\\]for some constant $c>0$.\n\nStraus has observed that if $2^k\\geq n$ then there exist $n$ points in $\\mathbb{R}^k$ which contain no isosceles triangle and determine at most $n-1$ distances.\n\nSee also [135].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 15 October 2025. View history", "references": "#657: [Er73][Er75f,p.101][ErPa90][Er97e]" }, { "number": 660, "url": "https://www.erdosproblems.com/660", "status": "open", "prize": "no", "tags": [ "geometry", "distances", "convex" ], "oeis": [], "formalized": "no", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\\[(1-o(1))\\frac{n}{2}\\]many distinct distances between the $x_i$?", "commentary": "For the similar problem in $\\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman [Al63] (see [93]). In [Er75f] Erdős claims that Altman proved that the vertices determine $\\gg n$ many distinct distances, but gives no reference.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 January 2026. View history", "references": "#660: [Er97e,p.531]" }, { "number": 661, "url": "https://www.erdosproblems.com/661", "status": "open", "prize": "$50", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Are there, for all large $n$, some points $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is\\[o\\left(\\frac{n}{\\sqrt{\\log n}}\\right)?\\]", "commentary": "One can also ask this for points in $\\mathbb{R}^3$. In $\\mathbb{R}^4$ Lenz observed that there are $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.\n\nMore generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minimal number of distinct distances between any $2n$ points in $\\mathbb{R}^2$, then is $F =o(f)$?\n\nSee also [89].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 January 2026. View history", "references": "#661: [ErPa90][Er92e][Er97e][Er97f]" }, { "number": 662, "url": "https://www.erdosproblems.com/662", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\\leq t$. For example $f(1)=6$, $f(\\sqrt{3})=12$, and $f(3)=18$.Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that $d(x_i,x_j)\\geq 1$ for all $i\\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?In particular, is it true that the number of distances $\\leq \\sqrt{3}-\\epsilon$ is less than $1$?", "commentary": "A problem of Erdős, Lovász, and Vesztergombi.\n\nThis is essentially verbatim the problem description in [Er97e], but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.\n\nErdős also goes on to write 'Perhaps the following stronger conjecture holds: Let $t_10$ and, for all large $n$, a pairwise balanced design such that\\[\\lvert A_i\\rvert > n^{1/2}-C\\]for all $1\\leq i\\leq m$?", "commentary": "A problem of Erdős and Larson [ErLa82]. In general, as Erdős asks in [Er97f], find the slowest growing function $h$ such that, for all large $n$, there exists a pairwise balanced design with\\[\\lvert A_i\\rvert > n^{1/2}-h(n)\\]for all $1\\leq i\\leq m$.\n\nThe problem above asks whether $h(n)\\ll 1$. Erdős and Larson prove that $h(n) \\ll n^{1/2-c}$ for some constant $c>0$, and note this can be improved to $h(n)\\ll (\\log n)^2$ assuming Cramer-type bounds on the difference between consecutive primes.\n\nShrikhande and Singhi [ShSi85] have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power (see [723]), by proving that every pairwise balanced design on $n$ points in which each block is of size $\\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\\leq c+2$, if $n$ is sufficiently large.\n\nIn general, if $H(n)$ is the largest prime gap $\\leq n$, then the above reuslts show that, assuming the prime power conjecture, $H(n)\\asymp h(n)$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#665: [ErLa82][Er97f,p.3]" }, { "number": 667, "url": "https://www.erdosproblems.com/667", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.Is\\[c(p,q)=\\liminf \\frac{\\log H(n)}{\\log n}\\]a strictly increasing function of $q$ for $1\\leq q\\leq \\binom{p-1}{2}+1$?", "commentary": "A problem of Erdős, Faudree, Rousseau, and Schelp.\n\nWhen $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example\\[\\frac{1}{p-1}\\leq c(p,1) \\leq \\frac{2}{p+1}.\\]It is easy to see that if $q=\\binom{p-1}{2}+1$ then $c(p,q)=1$. Erdős, Faudree, Rousseau, and Schelp have shown that $c(p,\\binom{p-1}{2})\\leq 1/2$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#667: [Er97f]" }, { "number": 668, "url": "https://www.erdosproblems.com/668", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Is it true that the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\\to\\infty$? Is it always $>1$ for $n>3$?", "commentary": "In fact this is $=1$ also for $n=4$, the unique example given by two equilateral triangles joined by an edge. \n\nComputational evidence of Engel, Hammond-Lee, Su, Varga, and Zsámboki [EHSVZ25] and Alexeev, Mixon, and Parshall [AMP25] suggests that this count is $=1$ for various other $5\\leq n\\leq 21$ (although these calculations were checking only up to graph isomorphism, rather than congruency). \n\nThe actual maximal number of unit distances is the subject of [90].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#668: [Er97f]" }, { "number": 669, "url": "https://www.erdosproblems.com/669", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [ "A003035", "A006065", "A008997" ], "formalized": "no", "statement": "Let $F_k(n)$ be minimal such that for any $n$ points in $\\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\\lim F_k(n)/n^2$ and $\\lim f_k(n)/n^2$.", "commentary": "Trivially $f_k(n)\\leq F_k(n)$ and $f_2(n)=F_2(n)=\\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Grünbaum, and Sloane [BGS74] have proved that\\[f_3(n)=\\frac{n^2}{6}-O(n)\\]and\\[F_3(n)=\\frac{n^2}{6}-O(n).\\]There is a trivial upper bound of $F_k(n) \\leq \\binom{n}{2}/\\binom{k}{2}$, and hence\\[\\lim F_k(n)/n^2 \\leq \\frac{1}{k(k-1)}.\\]See also [101].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#669: [Er97f]" }, { "number": 670, "url": "https://www.erdosproblems.com/670", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?", "commentary": "The lower bound of $\\binom{n}{2}$ for the diameter is trivial. Erdős [Er97f] proved the claim when $d=1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#670: [Er97f]" }, { "number": 671, "url": "https://www.erdosproblems.com/671", "status": "open", "prize": "$250", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Given $a_{i}^n\\in [-1,1]$ for all $1\\leq i\\leq n<\\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\\leq i'\\leq n$ with $i\\neq i'$. We similarly define\\[\\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i^n)p_i^n(x),\\]the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\\leq i\\leq n$ (that is, the sequence of Lagrange interpolation polynomials).Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists some $x\\in [-1,1]$ where\\[\\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty\\]and yet\\[\\mathcal{L}^nf(x) \\to f(x)?\\]Is there such a sequence such that\\[\\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty\\]for every $x\\in [-1,1]$ and yet for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists $x\\in [-1,1]$ with\\[\\mathcal{L}^nf(x) \\to f(x)?\\]", "commentary": "Bernstein [Be31] proved that for any choice of $a_i^n$ there exists $x_0\\in [-1,1]$ such that\\[\\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty.\\]Erdős and Vértesi [ErVe80] proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\\to \\mathbb{R}$ such that\\[\\limsup_{n\\to \\infty} \\lvert \\mathcal{L}^nf(x)\\rvert=\\infty\\]for almost all $x\\in [-1,1]$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#671: [Er82e][Er97f][Va99,2.40]" }, { "number": 672, "url": "https://www.erdosproblems.com/672", "status": "verifiable", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Can the product of an arithmetic progression of positive integers $n,n+d,\\ldots,n+(k-1)d$ of length $k\\geq 4$ (with $(n,d)=1$) be a perfect power?", "commentary": "Erdős believed not. Erdős and Selfridge [ErSe75] proved that the product of consecutive integers is never a perfect power.\n\nThe theory of Pell equations implies that there are infinitely many pairs $n,d$ with $(n,d)=1$ such that $n(n+d)(n+2d)$ is a square.\n\nConsidering the question of whether the product of an arithmetic progression of length $k$ can be equal to an $\\ell$th power:\nEuler proved this is impossible when $k=4$ and $\\ell=2$,\nObláth [Ob51] proved this is impossible when $(k,l)=(5,2),(3,3),(3,4),(3,5)$.\nMarszalek [Ma85] proved that this is only possible for $k\\ll_d 1$, where $d$ is the common difference of the arithmetic progression.\nGyöry, Hajdu, and Saradha [GHS04] proved this is impossible for $4\\leq k\\leq 5$.\nBennett, Bruin, Györy, and Hajdu [BBGH06] proved this is impossible for $4\\leq k\\leq 11$, and also impossible for $k$ sufficiently large depending only on the number of prime divisors of $d$.\n Györy, Hajdu, and Pintér [GHP09] have proved this is impossible for $4\\leq k\\leq 34$.\n Bennett and Siksek [BeSi20] have proved that this is impossible for all $k$ sufficiently large and $\\ell > e^{10^k}$ prime.\nJakob Führer has observed this is possible for integers in general, for example $(-6)\\cdot(-1)\\cdot 4\\cdot 9=6^3$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#672: [Er97c][Va99,1.32]" }, { "number": 675, "url": "https://www.erdosproblems.com/675", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "We say that $A\\subset \\mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\\geq 1$ such that, for all $1\\leq a\\leq n$,\\[a\\in A\\quad\\textrm{ if and only if }\\quad a+t_n\\in A.\\]\nDoes the set of the sums of two squares have the translation property?\nIf we partition all primes into $P\\sqcup Q$, such that each set contains $\\gg x/\\log x$ many primes $\\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?\nIf $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\\exp(n^c)$ for some constant $c>0$?", "commentary": "Elementary sieve theory implies that the set of squarefree numbers has the translation property.\n\nMore generally, Brun's sieve can be used to prove that if $B\\subseteq \\mathbb{N}$ is a set of pairwise coprime integers with $\\sum_{b0$. Erdős [Er79] believed it is 'rather unlikely' that all large integers are of this form.\n\nWhat if the condition that $p$ is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many $n$ not of this form even without the condition that $p$ is prime. It should be true that the number of exceptions in $[1,x]$ is $1$? Erdős expects very few (and none when $l\\geq k$). \n\nThe only solutions Erdős knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$. \n\nIn [Er79d] Erdős conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\\geq n+k$ then $\\prod_{i\\leq k}(n+i)$ and $\\prod_{i\\leq k}(m+i)$ cannot have the same set of prime factors.\n\nSee also [678], [686], and [850].\n\nThis is discussed in problem B35 of Guy's collection [Gu04]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#677: [Er79][Er79d][ErGr80]" }, { "number": 679, "url": "https://www.erdosproblems.com/679", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\\[\\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k}\\]for all $k0$.\n\nSee also [248], [413], and [1203].\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#679: [Er79d]" }, { "number": 680, "url": "https://www.erdosproblems.com/680", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "yes", "statement": "Is it true that, for all sufficiently large $n$, there exists some $k$ such that\\[p(n+k)>k^2+1,\\]where $p(m)$ denotes the least prime factor of $m$?Can one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all $\\epsilon>0$, where $C_\\epsilon>0$ is some constant?", "commentary": "This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.\n\nErdős observed that Cramer's conjecture\\[\\limsup_{k\\to \\infty} \\frac{p_{k+1}-p_k}{(\\log k)^2}=1\\]implies that for all $\\epsilon>0$ and all sufficiently large $n$ there exists some $k$ such that\\[p(n+k)>e^{(1-\\epsilon)\\sqrt{k}}.\\]There is now evidence, however, that Cramer's conjecture is false; a more refined heuristic by Granville [Gr95] suggests this $\\limsup$ is $2e^{-\\gamma}\\approx 1.119\\cdots$, and so perhaps the $1+\\epsilon$ in the second question should be replaced by $2e^{-\\gamma}+\\epsilon$.\n\nSee also [681] and [682].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#680: [Er79d]" }, { "number": 681, "url": "https://www.erdosproblems.com/681", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A389680" ], "formalized": "yes", "statement": "Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and\\[p(n+k)>k^2,\\]where $p(m)$ is the least prime factor of $m$?", "commentary": "Related to questions of Erdős, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.\n\nSee also [680] and [682].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#681: [Er79d]" }, { "number": 683, "url": "https://www.erdosproblems.com/683", "status": "open", "prize": "no", "tags": [ "number theory", "primes", "binomial coefficients" ], "oeis": [ "A006530", "A074399", "A121359" ], "formalized": "yes", "statement": "Is it true that for every $1\\leq k\\leq n$ the largest prime divisor of $\\binom{n}{k}$, say $P(\\binom{n}{k})$, satisfies\\[P\\left(\\binom{n}{k}\\right)\\geq \\min(n-k+1, k^{1+c})\\]for some constant $c>0$?", "commentary": "A theorem of Sylvester and Schur (see [Er34]) states that $P(\\binom{n}{k})>k$ if $k\\leq n/2$. Erdős [Er55d] proved that there exists some $c>0$ such that, whenever $k\\leq n/2$,\\[P\\left(\\binom{n}{k}\\right)\\gg k\\log k.\\]Erdős [Er79d] writes it 'seems certain' that this holds for every $c>0$, with only a finite number of exceptions (depending on $c$). Standard heuristics on prime gaps suggest that the largest prime divisor of $\\binom{n}{k}$ is, for $k\\leq n/2$, in fact\\[>e^{c\\sqrt{k}}\\]for some constant $c>0$. \n\nThis is essentially equivalent to [961].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 December 2025. View history", "references": "#683: [Er76d][Er79d]" }, { "number": 684, "url": "https://www.erdosproblems.com/684", "status": "open", "prize": "no", "tags": [ "number theory", "primes", "binomial coefficients" ], "oeis": [ "A392019" ], "formalized": "no", "statement": "For $0\\leq k\\leq n$ write\\[\\binom{n}{k} = uv\\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.", "commentary": "A classical theorem of Mahler states that for any $\\epsilon>0$ and integers $k$ and $l$ then, writing\\[(n+1)\\cdots (n+k) = ab\\]where the only primes dividing $a$ are $\\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\\epsilon}$ for all sufficiently large (depending on $\\epsilon,k,l$) $n$.\n\nMahler's theorem implies $f(n)\\to \\infty$ as $n\\to \\infty$, but is ineffective, and so gives no bounds on the growth of $f(n)$.\n\nOne can similarly ask for estimates on the smallest integer $f(n,k)$ such that if $m$ is the factor of $\\binom{n}{k}$ containing all primes $\\leq f(n,k)$ then $m > n^2$.\n\nTang and ChatGPT have proved that\\[f(n)\\leq n^{30/43+o(1)}.\\]The same proof would prove $f(n) \\leq n^{2/3+o(1)}$ assuming the Riemann Hypothesis (or Density Hypothesis). \n\nA heuristic given by Sothanaphan and ChatGPT in the comments suggests that, at least for most $n$, $f(n)\\sim 2\\log n$.\n\nAn internal OpenAI model [APSSV26] has given an elementary argument which shows $f(n) \\ll (\\log n)^2$, and constructs arbitrarily large $n$ such that $f(n)\\geq (1/2-o(1))\\log n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#684: [Er79d]" }, { "number": 685, "url": "https://www.erdosproblems.com/685", "status": "open", "prize": "no", "tags": [ "number theory", "primes", "binomial coefficients" ], "oeis": [], "formalized": "no", "statement": "Let $\\epsilon>0$ and $n$ be large depending on $\\epsilon$. Is it true that for all $n^\\epsilon\\frac{\\log \\binom{n}{k}}{\\log n},\\]and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#685: [Er79d]" }, { "number": 686, "url": "https://www.erdosproblems.com/686", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Can every integer $N\\geq 2$ be written as\\[N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}\\]for some $k\\geq 2$ and $m\\geq n+k$?", "commentary": "If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?\n\nSee also [388] and [677].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#686: [Er79d]" }, { "number": 687, "url": "https://www.erdosproblems.com/687", "status": "open", "prize": "$1000", "tags": [ "number theory" ], "oeis": [ "A048670", "A058989" ], "formalized": "no", "statement": "Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\\pmod{p}$. Give good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\\ll x^{1+o(1)}$?", "commentary": "This function (associated with Jacobsthal) is closely related to the problem of gaps between primes (see [4]). The best known upper bound is due to Iwaniec [Iw78],\\[Y(x) \\ll x^2.\\]The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18],\\[Y(x) \\gg x\\frac{\\log x\\log\\log\\log x}{\\log\\log x},\\]improving on a previous bound of Rankin [Ra38]. \n\nMaier and Pomerance have conjectured that $Y(x)\\ll x(\\log x)^{2+o(1)}$. \n\nIn [Er80] he writes 'It is not clear who first formulated this problem - probably many of us did it independently. I offer the maximum of \\$1000 dollars and $1/2$ my total savings for clearing up of this problem.'\n\nIn [Er80] Erdős also asks about a weaker variant in which all except $o(y/\\log y)$ of the integers in $[1,y]$ are congruent to at least one of the $a_p\\pmod{p}$, and in particular asks if the answer is very different.\n\nSee also [688] and [689]. A more general Jacobsthal function is the focus of [970].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#687: [Er79d,p.79][Er80,p.106][Er96b]" }, { "number": 688, "url": "https://www.erdosproblems.com/688", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Define $\\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\\epsilon_n}\\alpha$.\n\nTenenbaum notes in [Te96] that this is certainly not true as written since if the $n_j$ grow sufficiently quickly then this sequence is never Behrend, for any choice of $\\eta_k$. He then writes 'we understand from subsequent discussions with Erdős that he had actually in mind a two-sided condition on' $n_{j+1}/n_j$.\n\nTenenbaum [Te96] proves this conjecture: if there are constants $1\\log 2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#691: [Er79e,p.77]" }, { "number": 693, "url": "https://www.erdosproblems.com/693", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [ "A391118" ], "formalized": "no", "statement": "Let $k\\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\\{a_1i$ which divides $\\binom{n}{i}$.\n\nErdős and Szekeres further conjectured that $p\\geq i$ can be improved to $p>i$ except in a few special cases. In particular this fails when $i=2$ and $n$ being some particular powers of $2$. They also found some counterexamples when $i=3$, but only one counterexample when $i\\geq 4$:\\[\\textrm{gcd}\\left(\\binom{28}{5},\\binom{28}{14}\\right)=2^3\\cdot 3^3\\cdot 5.\\]This is mentioned in problem B31 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#699: [ErSz78]" }, { "number": 700, "url": "https://www.erdosproblems.com/700", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A091963" ], "formalized": "no", "statement": "Let\\[f(n)=\\min_{1n^{1/2}$?\n Is it true that, for every composite $n$,\\[f(n) \\ll_A \\frac{n}{(\\log n)^A}\\]for every $A>0$?", "commentary": "A problem of Erdős and Szekeres. It is easy to see that $f(n)\\leq n/P(n)$ for composite $n$, since if $j=p^k$ where $p^k\\mid n$ and $p^{k+1}\\nmid n$ then $\\textrm{gcd}\\left(n,\\binom{n}{j}\\right)=n/p^k$. This implies\\[f(n) \\leq (1+o(1))\\frac{n}{\\log n}.\\]It is known that $f(n)=n/P(n)$ when $n$ is the product of two primes. Another example is $n=30$.\n\nFor the second problem, it is easy to see that for any $n$ we have $f(n)\\geq p(n)$, where $p(n)$ is the smallest prime dividing $n$, and hence there are infinitely many $n$ (those $=p^2)$ such that $f(n)\\geq n^{1/2}$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#700: [ErSz78]" }, { "number": 701, "url": "https://www.erdosproblems.com/701", "status": "open", "prize": "no", "tags": [ "combinatorics", "intersecting family" ], "oeis": [], "formalized": "no", "statement": "Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that whenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have\\[\\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert.\\]", "commentary": "A problem of Chvátal [Ch74], who proved it replacing the closed under subsets condition with the (stronger) condition that, assuming all sets in $\\mathcal{F}$ are subsets of $\\{1,\\ldots,n\\}$, whenever $A\\in \\mathcal{F}$ and there is an injection $f:B\\to A$ such that $x\\leq f(x)$ for all $x\\in B$, then $B\\in \\mathcal{F}$.\n\nSterboul [St74] proved this when, letting $\\mathcal{G}$ be the maximal sets (under inclusion) in $\\mathcal{F}$, all sets in $\\mathcal{G}$ have the same size, $\\lvert A\\cap B\\rvert\\leq 1$ for all $A\\neq B\\in \\mathcal{G}$, and at least two sets in $\\mathcal{G}$ have non-empty intersection.\n\nFrankl and Kupavskii [FrKu23] have proved this when $\\mathcal{F}$ has covering number $2$.\n\nBorg [Bo11] has proposed a weighted generalisation of this conjecture, which he proves under certain additional assumptions.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#701: [Er81,p.26]" }, { "number": 704, "url": "https://www.erdosproblems.com/704", "status": "open", "prize": "no", "tags": [ "graph theory", "geometry", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $G_n$ be the unit distance graph in $\\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.Estimate the chromatic number $\\chi(G_n)$. Does it grow exponentially in $n$? Does\\[\\lim_{n\\to \\infty}\\chi(G_n)^{1/n}\\]exist?", "commentary": "A generalisation of the Hadwiger-Nelson problem (which addresses $n=2$). Frankl and Wilson [FrWi81] proved exponential growth:\\[\\chi(G_n) \\geq (1+o(1))1.2^n.\\]This was improved to\\[\\chi(G_n)\\geq (1.239\\cdots+o(1))^n\\]by Raigorodsky [Ra00]. The trivial colouring (by tiling with cubes) gives\\[\\chi(G_n) \\leq (2+\\sqrt{n})^n.\\]Larman and Rogers [LaRo72] improved this to\\[\\chi(G_n) \\leq (3+o(1))^n,\\]and conjecture the truth may be $(2^{3/2}+o(1))^n$. (Note that $2^{3/2}\\approx 2.828$.) Prosanov [Pr20] has given an alternative proof of this upper bound.\n\nSee also [508], [705], and [706].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#704: [Er81]" }, { "number": 706, "url": "https://www.erdosproblems.com/706", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $\\mathbb{R}^2$ and some set $A\\subset (0,\\infty)$ of size $r$, where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$, then $\\chi(G)\\leq L(r)$.Estimate $L(r)$. In particular, is it true that $L(r)\\leq r^{O(1)}$?", "commentary": "The case $r=1$ is the Hadwiger-Nelson problem, for which it is known that $5\\leq L(1)\\leq 7$.\n\n\nSee also [508], [704], and [705].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#706: [Er81]" }, { "number": 708, "url": "https://www.erdosproblems.com/708", "status": "open", "prize": "$100", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $g(n)$ be minimal such that for any $A\\subseteq [2,\\infty)\\cap \\mathbb{N}$ with $\\lvert A\\rvert =n$ and any set $I$ of $\\max(A)$ consecutive integers there exists some $B\\subseteq I$ with $\\lvert B\\rvert=g(n)$ such that\\[\\prod_{a\\in A} a \\mid \\prod_{b\\in B}b.\\]Is it true that\\[g(n) \\leq (2+o(1))n?\\]Or perhaps even $g(n)\\leq 2n$?", "commentary": "A problem of Erdős and Surányi [ErSu59], who proved that $g(n) \\geq (2-o(1))n$, and that $g(3)=4$. Their lower bound construction takes $A$ as the set of $p_ip_j$ for $i\\neq j$, where $p_1<\\cdots p_\\ell^2$.\n\nGallai was the first to consider problems of this type, and observed that $g(2)=2$ and $g(3)\\geq 4$. \n\nIn [Er92c] Erdős offers '100 dollars or 1000 rupees', whichever is more, for a proof or disproof. (In 1992 1000 rupees was worth approximately \\$38.60.)\n\nErdős and Surányi similarly asked what is the smallest $c_n\\geq 1$ such that in any interval $I\\subset [0,\\infty)$ of length $c_n\\max(A)$ there exists some $B\\subseteq I\\cap \\mathbb{N}$ with $\\lvert B\\rvert=n$ such that\\[\\prod_{a\\in A} a \\mid \\prod_{b\\in B}b.\\]They prove $c_2=1$ and $c_3=\\sqrt{2}$, but have no good upper or lower bounds in general.\n\n\nSee also [709].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#708: [ErSu59][Er92c][Er92e]" }, { "number": 709, "url": "https://www.erdosproblems.com/709", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be minimal such that, for any $A=\\{a_1,\\ldots,a_n\\}\\subseteq [2,\\infty)\\cap\\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\\max(A)$ consecutive integers there exist distinct $x_1,\\ldots,x_n\\in I$ such that $a_i\\mid x_i$.Obtain good bounds for $f(n)$, or even an asymptotic formula.", "commentary": "A problem of Erdős and Surányi [ErSu59], who proved\\[(\\log n)^c \\ll f(n) \\ll n^{1/2}\\]for some constant $c>0$. The lower bound here can be improved, using van Doorn's lower bound for [711] from [vD26], to\\[\\frac{\\log n}{\\log\\log n}\\ll f(n).\\]See also [708].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#709: [ErSu59][Er92c]" }, { "number": 710, "url": "https://www.erdosproblems.com/710", "status": "open", "prize": "₹2000", "tags": [ "number theory" ], "oeis": [ "A390246" ], "formalized": "no", "statement": "Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Obtain an asymptotic formula for $f(n)$.", "commentary": "A problem of Erdős and Pomerance [ErPo80], who proved\\[(2/\\sqrt{e}+o(1))n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}\\leq f(n)\\leq (1.7398\\cdots+o(1))n(\\log n)^{1/2}.\\]In [Er92c] Erdős offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\n\nSee also [711].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#710: [ErPo80][Er92c]" }, { "number": 711, "url": "https://www.erdosproblems.com/711", "status": "open", "prize": "₹1000", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Prove that\\[\\max_m f(n,m) \\leq n^{1+o(1)}\\]and that\\[\\max_m (f(n,m)-f(n,n))\\to \\infty.\\]", "commentary": "A problem of Erdős and Pomerance [ErPo80], who proved that\\[\\max_m f(n,m) \\ll n^{3/2}\\]and\\[n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2} \\ll f(n,n)\\ll n(\\log n)^{1/2}.\\]In [Er92c] Erdős offered 1000 rupees for a proof of either; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\n\nvan Doorn [vD26] has provided an affirmative answer to the second question, proving that, for all large $n$, there exists $m=m(n)$ such that\\[f(n,m)-f(n,n) \\gg \\frac{\\log n}{\\log\\log n}n.\\]See also [710].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 January 2026. View history", "references": "#711: [ErPo80][Er92c,p.36]" }, { "number": 712, "url": "https://www.erdosproblems.com/712", "status": "open", "prize": "$500", "tags": [ "graph theory", "turan number", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Determine, for any $k>r>2$, the value of\\[\\frac{\\mathrm{ex}_r(n,K_k^r)}{\\binom{n}{r}},\\]where $\\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $\\binom{k}{r}$ possible $r$-edges.", "commentary": "Turán proved that, when $r=2$, this limit is\\[\\frac{1}{2}\\left(1-\\frac{1}{k-1}\\right).\\]Erdős [Er81] offered \\$500 for the determination of this value for any fixed $k>r>2$, and \\$1000 for 'clearing up the whole set of problems'.\n\nSee also [500] for the case $r=3$ and $k=4$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 05 October 2025. View history", "references": "#712: [Er71,p.104][Er74c,p.76][Er81]" }, { "number": 713, "url": "https://www.erdosproblems.com/713", "status": "open", "prize": "$500", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Is it true that, for every bipartite graph $G$, there exists some $\\alpha\\in [1,2)$ and $c>0$ such that\\[\\mathrm{ex}(n;G)\\sim cn^\\alpha?\\]Must $\\alpha$ be rational?", "commentary": "A problem of Erdős and Simonovits. Erdős sometimes asked this in the weaker version with just\\[\\mathrm{ex}(n;G)\\asymp n^{\\alpha}.\\]Erdős [Er67d] had initially conjectured that, for any bipartite graph $G$, $\\mathrm{ex}(n;G)\\sim cn^{\\alpha}$ for some constant $c>0$ and $\\alpha$ of the shape $1+\\frac{1}{k}$ or $2-\\frac{1}{k}$ for some integer $k\\geq 2$. This was disproved by Erdős and Simonovits [ErSi70].\n\nThe analogous statement is not true for hypergraphs, as shown by Frankl and Füredi [FrFu87], who proved that if $G$ is the $5$-uniform hypergraph on $8$ vertices with edges $\\{12346,12457,12358\\}$ then $\\mathrm{ex}(n;G)=o(n^5)$ but $\\mathrm{ex}(n;G)\\neq O(n^c)$ for any $c<5$.\n\nA simplified proof was given by Füredi and Gerbner [FuGe21], who extended it to a counterexample for all $k\\geq 5$. It remains open whether it is true for $k=3$ and $k=4$ (though Füredi and Gerbner conjecture it is not).\n\nSee also [571].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#713: [ErSi70,p.379][Er74c,p.78][Er75][Er78,p.30][Er81,p.30][ErSi84][Er91]" }, { "number": 714, "url": "https://www.erdosproblems.com/714", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Is it true that\\[\\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}?\\]", "commentary": "Kövári, Sós, and Turán [KST54] proved\\[\\mathrm{ex}(n; K_{r,r}) \\ll n^{2-1/r}\\]for all $r\\geq 2$. Brown [Br66] and, independently, Erdős, Rényi, and Sós [ERS66], proved the conjectured lower bound when $r=3$.\n\nWhen $r=2$ it is known that\\[\\mathrm{ex}(n;K_{2,2})=\\left(\\frac{1}{2}+o(1)\\right)n^{3/2}\\](see [768], since $K_{2,2}=C_4$).\n\nSee also [147]. The generalisation to hypergraphs is [1158].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#714: [Er64c][Er67b][Er69][Er71,p.103][Er74c,p.77][Er75][Er81][Er93,p.334]" }, { "number": 719, "url": "https://www.erdosproblems.com/719", "status": "open", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $\\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).Is every $r$-hypergraph $G$ on $n$ vertices the union of at most $\\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?", "commentary": "A conjecture of Erdős and Sauer.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#719: [Er81]" }, { "number": 723, "url": "https://www.erdosproblems.com/723", "status": "falsifiable", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "yes", "statement": "If there is a finite projective plane of order $n$ then must $n$ be a prime power?A finite projective plane of order $n$ is a collection of subsets of $\\{1,\\ldots,n^2+n+1\\}$ of size $n+1$ such that every pair of elements is contained in exactly one set.", "commentary": "These always exist if $n$ is a prime power. This conjecture has been proved for $n\\leq 11$, but it is open whether there exists a projective plane of order $12$.\n\nBruck and Ryser [BrRy49] have proved that if $n\\equiv 1\\pmod{4}$ or $n\\equiv 2\\pmod{4}$ then $n$ must be the sum of two squares. For example, this rules out $n=6$ or $n=14$. The case $n=10$ was ruled out by computer search [La97].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#723: [Er81]" }, { "number": 724, "url": "https://www.erdosproblems.com/724", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [ "A001438" ], "formalized": "no", "statement": "Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that\\[f(n) \\gg n^{1/2}?\\]", "commentary": "Euler conjectured that $f(n)=1$ when $n\\equiv 2\\pmod{4}$, but this was disproved by Bose, Parker, and Shrikhande [BPS60] who proved $f(n)\\geq 2$ for $n\\geq 7$.\n\nChowla, Erdős, and Straus [CES60] proved $f(n) \\gg n^{1/91}$. Wilson [Wi74] proved $f(n) \\gg n^{1/17}$. Beth [Be83c] proved $f(n) \\gg n^{1/14.8}$. \n\nThe sequence of $f(n)$ is A001438 in the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#724: [Er81]" }, { "number": 725, "url": "https://www.erdosproblems.com/725", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [ "A001009" ], "formalized": "no", "statement": "Give an asymptotic formula for the number of $k\\times n$ Latin rectangles.", "commentary": "Erdős and Kaplansky [ErKa46] proved the count is\\[\\sim e^{-\\binom{k}{2}}(n!)^k\\]when $k=o((\\log n)^{3/2-\\epsilon})$. Yamamoto [Ya51] extended this to $k\\leq n^{1/3-o(1)}$.\n\nThe count of such Latin rectangles is A001009 in the OEIS.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#725: [Er81]" }, { "number": 726, "url": "https://www.erdosproblems.com/726", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "As $n\\to \\infty$ ranges over integers\\[\\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}.\\]", "commentary": "A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75]. For comparison the classical estimate of Mertens states that\\[\\sum_{p\\leq n}\\frac{1}{p}\\sim \\log\\log n.\\]By $n\\in (p/2,p)\\pmod{p}$ we mean $n\\equiv r\\pmod{p}$ for some integer $r$ with $p/20$). \n\nErdős [Er68c] proved that if $a!b!\\mid n!$ then $a+b\\leq n+O(\\log n)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#727: [EGRS75]" }, { "number": 730, "url": "https://www.erdosproblems.com/730", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients", "base representations" ], "oeis": [ "A129515" ], "formalized": "yes", "statement": "Are there infinitely many pairs of integers $n\\neq m$ such that $\\binom{2n}{n}$ and $\\binom{2m}{m}$ have the same set of prime divisors?", "commentary": "A problem of Erdős, Graham, Ruzsa, and Straus [EGRS75], who believed there is 'no doubt' that the answer is yes. \n\nFor example $(87,88)$ and $(607,608)$. Those $n$ such that there exists some suitable $m>n$ are listed as A129515 in the OEIS. \n\nA triple of such $n$ for which $\\binom{2n}{n}$ all share the same set of prime divisors is $(10003,10004,10005)$. It is not known whether there are such pairs of the shape $(n,n+k)$ for every $k\\geq 1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#730: [EGRS75]" }, { "number": 731, "url": "https://www.erdosproblems.com/731", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A006197" ], "formalized": "no", "statement": "Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\\nmid \\binom{2n}{n}$ satisfies\\[m\\sim f(n).\\]", "commentary": "A problem of Erdős, Graham, Ruzsa, and Straus [EGRS75], who say it is 'not hard to show that', for almost all $n$, the minimal such $m$ satisfies\\[m=\\exp((\\log n)^{1/2+o(1)}).\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#731: [EGRS75]" }, { "number": 734, "url": "https://www.erdosproblems.com/734", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "no", "statement": "Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\\lvert A_i\\rvert=t$.", "commentary": "$A_1,\\ldots,A_m$ is a pairwise balanced block design if every pair in $\\{1,\\ldots,n\\}$ is contained in exactly one of the $A_i$. \n\nErdős [Er81] writes 'this will be probably not be very difficult to prove but so far I was not successful'.\n\nErdős and de Bruijn [dBEr48] proved that if $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ is a pairwise balanced block design then $m\\geq n$, and this implies there must be some $t$ such that there are $\\gg n^{1/2}$ many $t$ with $\\lvert A_i\\rvert=t$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#734: [Er81,p.35]" }, { "number": 736, "url": "https://www.erdosproblems.com/736", "status": "not provable", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph with chromatic number $\\aleph_1$. Is there, for every cardinal number $m$, some graph $G_m$ of chromatic number $m$ such that every finite subgraph of $G_m$ is a subgraph of $G$?", "commentary": "A conjecture of Walter Taylor. The more general problem replaces $\\aleph_1$ with any uncountable cardinal $\\kappa$.\n\nMore generally, Erdős asks to characterise families $\\mathcal{F}_\\alpha$ of finite graphs such that there is a graph of chromatic number $\\aleph_\\alpha$ all of whose finite subgraphs are in $\\mathcal{F}_\\alpha$.\n\nKomjáth [KoSh05] proved that it is consistent that the answer is no, in that there exists a graph $G$ with chromatic number $\\aleph_1$ such that if $H$ is any graph all of whose finite subgraphs are subgraphs of $G$ then $H$ has chromatic number $\\leq \\aleph_2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#736: [Er81][Er93,p.343]" }, { "number": 738, "url": "https://www.erdosproblems.com/738", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?", "commentary": "A conjecture of Gyárfás.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#738: [Er81]" }, { "number": 739, "url": "https://www.erdosproblems.com/739", "status": "not provable", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Is it true that, for every infinite cardinal $\\mathfrak{n}< \\mathfrak{m}$, there exists a subgraph of $G$ with chromatic number $\\mathfrak{n}$?", "commentary": "A question of Galvin [Ga73], who proved that this is true with $\\mathfrak{m}=\\aleph_0$. Galvin also proved that the stronger version of this statement, in which the subgraph is induced, implies $2^{\\mathfrak{k}}<2^{\\mathfrak{n}}$ for all cardinals $\\mathfrak{k}<\\mathfrak{n}$. \n\nKomjáth [Ko88b] proved it is consistent that\\[2^{\\aleph_0}=2^{\\aleph_1}=2^{\\aleph_2}=\\aleph_3\\]and that there exist graphs which fail this property (with $\\mathfrak{m}=\\aleph_2$ and $\\mathfrak{n}=\\aleph_1$).\n\nShelah [Sh90] proved that, assuming the axiom of constructibility, the answer is yes with $\\mathfrak{m}=\\aleph_2$ and $\\mathfrak{n}=\\aleph_1$.\n\nIt remains open whether the answer to this question is yes assuming the Generalized Continuum Hypothesis, for example.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#739: [Er81]" }, { "number": 740, "url": "https://www.erdosproblems.com/740", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Let $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain any odd cycle of length $\\leq r$?", "commentary": "A question of Erdős and Hajnal. Rödl proved this is true if $\\mathfrak{m}=\\aleph_0$ and $r=3$ (see [108] for the finitary version).\n\nMore generally, Erdős and Hajnal asked must there exist (for every cardinal $\\mathfrak{m}$ and integer $r$) some $f_r(\\mathfrak{m})$ such that every graph with chromatic number $\\geq f_r(\\mathfrak{m})$ contains a subgraph with chromatic number $\\mathfrak{m}$ with no odd cycle of length $\\leq r$?\n\nErdős [Er95d] claimed that even the $r=3$ case of this is open: must every graph with sufficiently large chromatic number contain a triangle free graph with chromatic number $\\mathfrak{m}$?\n\nIn [Er81] Erdős also asks the same question but with girth (i.e. the subgraph does not contain any cycle at all of length $\\leq C$).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#740: [Er69b][Er71,p.100][Er81][Er95d]" }, { "number": 741, "url": "https://www.erdosproblems.com/741", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive (upper) density?Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", "commentary": "A problem of Burr and Erdős. Erdős [Er94b] thought he could construct a basis as in the second question, but 'could never quite finish the proof'.\n\nIn [Er94b] the first question is simply asked with 'positive density', but by this Erdős often meant positive upper and/or lower density, rather than necessarily that the density exists and is positive. The interpretation of this question with upper density most likely captures the spirit of the question.\n\nDeepMind and an internal OpenAI model (see the comments and [APSSV26]) have independently constructed a basis which gives an affirmative answer to the second question. DeepMind gave a counterexample to the strict density version of the first question.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#741: [Er94b,p.263]" }, { "number": 742, "url": "https://www.erdosproblems.com/742", "status": "decidable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the diameter of $G$. Is it true that $G$ has at most $n^2/4$ edges?", "commentary": "A conjecture of Murty and Plesnik (see [CaHa79]) (although Füredi credits this conjecture to Murty and Simon, and further mentions that Erdős told him that the conjecture goes back to Ore in the 1960s). The complete bipartite graph shows that this would be best possible.\n\nThis is true (for sufficiently large $n$) and was proved by Füredi [Fu92].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#742: [Er81]" }, { "number": 743, "url": "https://www.erdosproblems.com/743", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $T_2,\\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$?", "commentary": "A conjecture of Gyárfás, known as the tree packing conjecture.\n\nGyárfás and Lehel [GyLe78] proved that this holds if all but at most $2$ of the trees are stars, or if all the trees are stars or paths. Fishburn [Fi83] proved this for $n\\leq 9$. Bollobás [Bo83] proved that the smallest $\\lfloor n/\\sqrt{2}\\rfloor$ many trees can always be packed greedily into $K_n$.\n\nJoos, Kim, Kühn, and Osthus [JKKO19] proved that this conjecture holds when the trees have bounded maximum degree. Allen, Böttcher, Clemens, Hladky, Piguet, and Taraz [ABCHPT21] proved that this conjecture holds when all the trees have maximum degree $\\leq c\\frac{n}{\\log n}$ for some constant $c>0$.\n\nJanzer and Montgomery [JaMo24] have proved that there exists some $c>0$ such that the largest $cn$ trees can be packed into $K_n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#743: [Er81]" }, { "number": 749, "url": "https://www.erdosproblems.com/749", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?", "commentary": "It is unclear from [Er94b] whether Erdős believed the answer to this should be yes or no. He also asked a similar question for upper density, for which he seemed more certain the answer should be yes.\n\nHe wrote that both he and Sárközy believed that if $1_A\\ast 1-A(n)\\leq C$ for all $n$ then the upper density of $A+A$ is at most $1-\\epsilon_C$, where $\\epsilon_C>0$ may depend on $C$. \n\nThe upper density variant has been resolved by Bhalla, with the assistance of GPT 5.4. Bhalla constructs, for any $\\epsilon>0$, a set $A$ for which the upper density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n)\\ll \\epsilon^{-1}$ for all $n$.\n\nSee also [28].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#749: [Er94b,p.263]" }, { "number": 750, "url": "https://www.erdosproblems.com/750", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a graph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains an independent set of size at least $\\frac{m}{2}-f(m)$?", "commentary": "In [Er69b] Erdős conjectures this for $f(m)=\\epsilon m$ for any fixed $\\epsilon>0$. This follows from a result of Erdős, Hajnal, and Szemerédi [EHS82], as described by msellke in the comments.\n\nIn [ErHa67b] Erdős and Hajnal prove this for $f(m)\\geq cm$ for all $c>1/4$.\n\nSee also [75].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 October 2025. View history", "references": "#750: [Er94b][Er95d]" }, { "number": 757, "url": "https://www.erdosproblems.com/757", "status": "open", "prize": "no", "tags": [ "geometry", "distances", "sidon sets" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset \\mathbb{R}$ be a set of size $n$ such that every subset $B\\subseteq A$ with $\\lvert B\\rvert =4$ has $\\lvert B-B\\rvert\\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\\geq cn$.", "commentary": "For comparison, note that if $B$ were a Sidon set then $\\lvert B-B\\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.\n\nWithout loss of generality, one can assume $A\\subset \\mathbb{N}$. \n\nErdős and Sós proved that $c\\geq 1/2$. Gyárfás and Lehel [GyLe95] proved\\[\\frac{1}{2}+\\frac{1}{141\\cdot 76}\\leq c\\leq\\frac{3}{5}.\\](The example proving the upper bound is the set of the first $n$ Fibonacci numbers.)\nMa and Tang [MaTa26] have improved these bounds to\\[\\frac{9}{17}\\leq c\\leq \\frac{4}{7}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#757: [Er97b]" }, { "number": 761, "url": "https://www.erdosproblems.com/761", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. The dichromatic number of $G$, denoted by $\\delta(G)$, is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$, there is a $k$-colouring of the vertices of $G$ such that there are no monochromatic oriented cycles. Must a graph with large chromatic number have large dichromatic number? Must a graph with large cochromatic number contain a graph with large dichromatic number?", "commentary": "The first question is due to Erdős and Neumann-Lara. The second question is due to Erdős and Gimbel. A positive answer to the second question implies a positive answer to the first via the bound mentioned in [760].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#761: [ErGi93]" }, { "number": 766, "url": "https://www.erdosproblems.com/766", "status": "open", "prize": "no", "tags": [ "graph theory", "turan number" ], "oeis": [], "formalized": "no", "statement": "Let $f(n;k,l)=\\min \\mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges.Give good estimates for $f(n;k,l)$ in the range $k1$ such that $d\\equiv 1\\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$\\[\\frac{\\lvert A\\cap [1,N]\\rvert}{N}=\\exp(-(c+o(1))\\sqrt{\\log N}\\log\\log N).\\]", "commentary": "Erdős could prove that there exists some constant $c>0$ such that for all large $N$\\[\\exp(-c\\sqrt{\\log N}\\log\\log N)\\leq \\frac{\\lvert A\\cap [1,N]\\rvert}{N}\\]and\\[\\frac{\\lvert A\\cap [1,N]\\rvert}{N}\\leq \\exp(-(1+o(1))\\sqrt{\\log N\\log\\log N}).\\]Erdős asked about this because $\\lvert A\\cap [1,N]\\rvert$ provides an upper bound for the number of integers $n\\leq N$ for which there is a non-cyclic simple group of order $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 September 2025. View history", "references": "#768: [Er74b]" }, { "number": 769, "url": "https://www.erdosproblems.com/769", "status": "open", "prize": "no", "tags": [ "number theory", "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \\gg n^n$?", "commentary": "A problem first investigated by Hadwiger, who proved the lower bound\\[c(n) \\geq 2^n+2^{n-1}.\\]It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erdős [Er74b] proved\\[c(n) \\ll n^{n+1}.\\]Erdős wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'\n\nHudelson [Hu98] proved that if $(2^n-1,3^n-1)=1$ then $c(n) < 6^n$, and in general $c(n) \\ll (2n)^{n-1}$. Connor and Marmorino [CoMa18] proved that\\[c(n) \\geq 2^{n+1}-1\\]for all $n\\geq 3$,\\[c(n) \\leq 1.8n^{n+1}\\]if $n+1$ is prime, and\\[c(n) \\leq e^2n^n\\]otherwise. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#769: [Er74b]" }, { "number": 770, "url": "https://www.erdosproblems.com/770", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A263647" ], "formalized": "yes", "statement": "Let $h(n)$ be minimal such that $2^n-1,3^n-1,\\ldots,h(n)^n-1$ are mutually coprime. Does, for every prime $p$, the density $\\delta_p$ of integers with $h(n)=p$ exist? Does $\\liminf h(n)=\\infty$? Is it true that if $p$ is the greatest prime such that $p-1\\mid n$ and $p>n^\\epsilon$ then $h(n)=p$?", "commentary": "It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$. \n\nIt is probably true that $h(n)=3$ for infinitely many $n$.\n\nSee also [820].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 September 2025. View history", "references": "#770: [Er74b]" }, { "number": 773, "url": "https://www.erdosproblems.com/773", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets", "squares" ], "oeis": [ "A390813" ], "formalized": "no", "statement": "What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?", "commentary": "A question of Alon and Erdős [AlEr85], who proved $\\lvert A\\rvert \\geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that\\[\\lvert A\\rvert \\ll \\frac{N}{(\\log N)^{1/4}},\\]since (as shown by Landau) the density of the sums of two squares decays like $(\\log N)^{-1/2}$. The lower bound was improved to\\[\\lvert A\\rvert \\gg N^{2/3}\\]by Lefmann and Thiele [LeTh95].\n\nIn [Er80] Erdős further defines $g(A)$ to be the maximal size of a Sidon set of $\\{a^2 : a\\in A\\}$, and asks whether $g(A)\\geq g(\\{1,\\ldots,N\\})$ where $N=\\lvert A\\rvert$. (See [530] for the linear analogue.)\n\nHe further asks the infinite analogue: is there an infinite set $A\\subset \\mathbb{N}$ such that $\\lvert A\\cap [1,N]\\rvert \\geq N^{1-o(1)}$ for all large $N$ and $\\{a^2 : a \\in A\\}$ is a Sidon set?\n\nSee [1206] for a similar question for higher powers.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#773: [Er80,p.109][AlEr85]" }, { "number": 774, "url": "https://www.erdosproblems.com/774", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\\neq \\sum_{m\\in Y}m$ for all finite $X,Y\\subset A$ with $X\\neq Y$. Let $A\\subset \\mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\\subset A$ contains a dissociated set of size $\\gg \\lvert B\\rvert$.Is every proportionately dissociated set the union of a finite number of dissociated sets?", "commentary": "This question appears in a paper of Alon and Erdős [AlEr85], although the general topic was first considered by Pisier [Pi83], who observed that the converse holds, and proved that being proportionately dissociated is equivalent to being a 'Sidon set' in the harmonic analysis sense; that is, whenever $f:A\\to \\mathbb{C}$ there exists some $\\theta\\in [0,1]$ such that\\[\\| f\\|_1 \\ll \\left\\lvert\\sum_{n\\in A} f(n)e(n\\theta)\\right\\rvert,\\]where $e(x)=e^{2\\pi ix}$. \n\nAlon and Erdős write that it 'seems unlikely that [this] is also sufficient'. They also point out the same question can be asked replacing dissociated with Sidon (in the additive combinatorial sense) (see [328]). This latter question was resolved in the negative by Nešetřil, Rödl, and Sales [NRS24].\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#774: [AlEr85][Er92b]" }, { "number": 776, "url": "https://www.erdosproblems.com/776", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 2$ and $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ be such that $A_i\\not\\subseteq A_j$ for all $i\\neq j$ and for any $t$ if there exists some $i$ with $\\lvert A_i\\rvert=t$ then there must exist at least $r$ sets of that size.How large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?", "commentary": "A problem of Erdős and Trotter. For $r=1$ and $n>3$ the maximum possible is $n-2$. For $r>1$ and $n$ sufficiently large $n-3$ is achievable, but $n-2$ is never achievable.\n\nLet $n_0(r)$ be such that whenever $n>n_0(r)$ there exists such a family with $n-3$ distinct set sizes. He and Tang [HeTa26b] (with ChatGPT) have proved that $n_0(2)=3$, $n_0(3)=8$, and for all $r\\geq 4$\\[2r+2\\leq n_0(r)\\leq 2r+2\\log_2r+O(\\log\\log r),\\]and hence $n_0(r)\\sim 2r$ as $r\\to \\infty$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#776: [Er81i][Gu83]" }, { "number": 778, "url": "https://www.erdosproblems.com/778", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques.Does Bob have a winning strategy for $n\\geq 3$? (Erdős believed the answer is yes.)If we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$?Finally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?", "commentary": "Malekshahian and Spiro [MaSp24] have proved that, for the first game, the set of $n$ for which Bob wins has density at least $3/4$ - in fact they prove that if Alice wins at $n$ then Bob wins at $n+1,n+2,n+3$. \n\nSimilarly, for the third game they prove that the set of $n$ for which Bob wins has density at least $2/3$, and prove the stronger statement that if Alice wins at $n$ then Bob wins at $n+1,n+2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#778: [Gu83]" }, { "number": 779, "url": "https://www.erdosproblems.com/779", "status": "falsifiable", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A005235" ], "formalized": "yes", "statement": "Let $n> 1$ and $p_1<\\cdots0$ such that, for any $k$, the squares contain a sequence $x_1,\\ldots,x_k$ where, for some $d$ and all $1\\leq i0$ and let $N$ be large. Let $A\\subseteq \\{2,\\ldots,N\\}$ be such that $(a,b)=1$ for all $a\\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$.What choice of such an $A$ minimises the number of integers $m\\leq N$ not divisible by any $a\\in A$?", "commentary": "Erdős [Er73] suggests that, if $p_i$ is the $i$th prime, then choosing $A=\\{p_r<\\cdots0$. Is there some set $A\\subset \\mathbb{N}$ of density $>1-\\epsilon$ such that $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_i,b_j\\in A$ can only hold when $r=s$?Similarly, can one always find a set $A\\subset\\{1,\\ldots,N\\}$ with this property of size $\\geq (1-o(1))N$?", "commentary": "An example of such a set with density $1/4$ is given by the integers $\\equiv 2\\pmod{4}$.\n\nSelfridge constructed such a set with density $1/e-\\epsilon$ for any $\\epsilon>0$: let $p_1<\\cdotsN^{1/2}$ give an example of a set with size $\\geq (\\log 2)N$. Erdős could improve this constant slightly. Such an improvement is given by Tao in the comments, taking $A$ to be the set of all integers with exactly one prime factor $>N^{\\frac{1}{1+\\sqrt{e}}}$, which has size $\\approx 0.8285 N$.\n\nIn [Er65] Erdős reports that Ruzsa proved the maximal size of such an $A$ is $\\leq (1-c)N$ for some constant $c>0$ for large $N$, but the proof 'is not yet published'. It is presumably the following.\n\nFor any such $A$ we can define an associated additive function $f$ by letting $f(a_1\\cdots a_r)=r$, and then $A=\\{ n: f(n)=1\\}$. A result of Erdős, Ruzsa, and Sárkzözy [ERS73] then implies that there exists a constant $c>0$ such that, for all sufficiently large $N$, $\\lvert A\\rvert\\leq (1-c)N$. In [ERS73] they indicate that their method should be able to obtain this with $c=1/10$. In the comments Tao explains how a result of Granville and Soundararajan [GrSo01] implies\\[\\lvert A\\rvert\\leq (1-c+o(1))N\\]for an explicit constant $c\\approx 0.1715$, and this is the best possible (see also [121]).\n\nTheorem 2 from [ERS73] also implies a negative answer to the first question, in that the density is at most $1/2$.\n\nThe above all assumes that repetition is allowed amongst the $a_i$ and $b_i$ in this condition. It is ambiguous in the original sources cited whether this was intended or not. If we insist the products are of distinct elements then this is a weaker condition on $A$, and the above questions are still open.\n\nThe formulation in [Er80] is clear that repetitions are not allowed, but also claims that Ruzsa had shown the answer to both questions is no in this case (and that Ruzsa had shown that the upper density of $A$ is $<1/e$ if $A$ is infinite with this property); it is possible that Erdős was mistakenly thinking of the above results, in which repetitions are allowed.\n\nSee also [421] and [795].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#786: [Er65][Er69,p.81][Er73,p.132][Er80,p.114]" }, { "number": 787, "url": "https://www.erdosproblems.com/787", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ such that $b_1+b_2\\not\\in A$ for all $b_1\\neq b_2\\in B$.Estimate $g(n)$.", "commentary": "This function was considered by Erdős and Moser. Choi observed that, without loss of generality, one can assume that $A\\subset \\mathbb{Z}$.\n\nKlarner proved $g(n) \\gg \\log n$ (indeed, a greedy construction suffices). Choi [Ch71] proved $g(n) \\ll n^{2/5+o(1)}$. The current best bounds known are\\[(\\log n)^{1+c} \\ll g(n) \\ll \\exp(\\sqrt{\\log n})\\]for some constant $c>0$, the lower bound due to Sanders [Sa21] and the upper bound due to Ruzsa [Ru05]. Beker [Be25] has proved\\[(\\log n)^{1+\\tfrac{1}{68}+o(1)} \\ll g(n).\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#787: [Er65,p.187][Er73,p.130][Va99,1.22]" }, { "number": 788, "url": "https://www.erdosproblems.com/788", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\\not\\in B$ for all $c_1\\neq c_2\\in C$ and $\\lvert C\\rvert+\\lvert B\\rvert \\geq f(n)$. Estimate $f(n)$. In particular is it true that $f(n)\\leq n^{1/2+o(1)}$?", "commentary": "A conjecture of Choi [Ch71], who proved $f(n) \\ll n^{3/4}$. Adenwalla in the comments has provided a simple construction that proves $f(n) \\gg n^{1/2}$.\n\nHunter in the comments has sketched an argument that gives $f(n) \\ll n^{2/3+o(1)}$. The bound\\[f(n) \\ll (n\\log n)^{2/3}\\]was proved by Baltz, Schoen, and Srivastav [BSS00].\n\nIn general, the argument given by Hunter shows that, if a random Cayley graph with probability $p$ has (almost surely) independence number $\\ll p^{-c-o(1)}$, then $f(n) \\leq n^{\\frac{c}{c+1}+o(1)}$.\n\nThe work of Alon and Pham [AlPh25] on random Cayley graphs therefore implies that\\[f(n) \\leq n^{3/5+o(1)},\\]and the conjectured bound of $\\leq p^{-1+o(1)}$ for the independence number would yield $f(n)\\leq n^{1/2+o(1)}$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 January 2026. View history", "references": "#788: [Er73]" }, { "number": 789, "url": "https://www.erdosproblems.com/789", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if $a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.Estimate $h(n)$.", "commentary": "Erdős [Er62c] proved $h(n) \\ll n^{5/6}$. Straus [St66] proved $h(n) \\ll n^{1/2}$. Erdős noted the bound $h(n)\\gg n^{1/3}$, taking\\[B=\\{ a: \\{ \\alpha a\\} \\in n^{-1/3}+\\tfrac{1}{2} (-n^{-2/3},n^{-2/3})\\}\\]for a random $\\alpha\\in [0,1]$. [Er62c] and Choi [Ch74b] improved this to $h(n) \\gg (n\\log n)^{1/3}$.\n\nSee also [186] and [874].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#789: [Er65][Er73]" }, { "number": 790, "url": "https://www.erdosproblems.com/790", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $l(n)$ be maximal such that if $A\\subset\\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there exists a sum-free $B\\subseteq A$ with $\\lvert B\\rvert \\geq l(n)$ - that is, $B$ is such that there are no solutions to\\[a_1=a_2+\\cdots+a_r\\]with $a_i\\in B$ all distinct.Estimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\\to \\infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?", "commentary": "Erdős observed that $l(n)\\geq (n/2)^{1/2}$, which Choi improved to $l(n)>(1+c)n^{1/2}$ for some $c>0$. Erdős [Er73] thought he could prove $l(n)=o(n)$ but had 'difficulties in reconstructing [his] proof'. (In [Er65] he wrote 'by complicated arguments we can show $l(n)=o(n)$'.)\n\nChoi, Komlós, and Szemerédi [CKS75] proved\\[\\left(\\frac{\\log n}{\\log\\log n}n\\right)^{1/2}\\ll l(n) \\ll \\frac{n}{\\log n}.\\]They further conjecture that $l(n)\\geq n^{1-o(1)}$. \n\nSee also [876].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#790: [Er65,p.188][Er73,p.130][Va99,1.22]" }, { "number": 791, "url": "https://www.erdosproblems.com/791", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [ "A066063" ], "formalized": "no", "statement": "Let $g(n)$ be minimal such that there exists $A\\subseteq \\{0,\\ldots,n\\}$ of size $g(n)$ with $\\{0,\\ldots,n\\}\\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\\sim 2n^{1/2}$?", "commentary": "Such a set is often called a finite additive $2$-basis. A problem of Rohrbach, who proved in [Ro37]\\[(2+c)n \\leq g(n)^2 \\leq 4n\\]for some small constant $c>0$. The current best-known bounds are\\[(2.181\\cdots+o(1))n\\leq g(n)^2 \\leq (3.458\\cdots+o(1))n.\\]The lower bound is due to Yu [Yu15], and the upper bound is due to Kohonen [Ko17]. (The disproof of $g(n)\\sim 2n^{1/2}$ was accomplished by Mrose [Mr79], who gave a construction implying $g(n)^2 \\leq \\frac{7}{2}n$.)\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 September 2025. View history", "references": "#791: [Er73]" }, { "number": 792, "url": "https://www.erdosproblems.com/792", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be maximal such that in any $A\\subset \\mathbb{Z}$ with $\\lvert A\\rvert=n$ there exists some sum-free subset $B\\subseteq A$ with $\\lvert B\\rvert \\geq f(n)$, so that there are no solutions to\\[a+b=c\\]with $a,b,c\\in B$. Estimate $f(n)$.", "commentary": "Erdős [Er65] gave a simple proof that shows $f(n) \\geq n/3$. Alon and Kleitman [AlKl90] improved this to $f(n)\\geq \\frac{n+1}{3}$, and Bourgain [Bo97] further improved this to $\\frac{n+2}{3}$. The best lower bound known is\\[f(n)\\geq \\frac{n}{3}+c\\log\\log n\\]for some constant $c>0$, due to Bedert [Be25b]. The best upper bound known is\\[f(n) \\leq \\frac{n}{3}+o(n),\\]due to Eberhard, Green, and Manners [EGM14].\n\nThis problem is Problem 1 on Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#792: [Er65][Er73][Er92c][Va99,1.22]" }, { "number": 793, "url": "https://www.erdosproblems.com/793", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,n\\}$ such that $a\\nmid bc$ whenever $a,b,c\\in A$ with $a\\neq b$ and $a\\neq c$. Is there a constant $C$ such that\\[F(n)=\\pi(n)+(C+o(1))n^{2/3}(\\log n)^{-2}?\\]", "commentary": "Erdős [Er38] proved there exist constants $0g(n)\\geq (\\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\\geq \\log n$ and an independent set of size $\\geq \\log n$?In particular, is there such a graph for $g(n)=(\\log n)^3$?", "commentary": "A problem of Erdős and Hajnal, who thought that there is no such graph for $g(n)=(\\log n)^3$. Alon and Sudakov [AlSu07] proved that there is no such graph with\\[g(n)=\\frac{c}{\\log\\log n}(\\log n)^3\\]for some constant $c>0$.\n\nAlon, Bucić, and Sudakov [ABS21] construct such a graph with\\[g(n)\\leq 2^{2^{(\\log\\log n)^{1/2+o(1)}}}.\\]See also [804].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#805: [Er91]" }, { "number": 809, "url": "https://www.erdosproblems.com/809", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Define the anti-Ramsey number $\\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours.Is it true that, for all $k\\geq 3$,\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\sim n^2/8?\\]", "commentary": "A problem of Burr, Erdős, Graham, and Sós [BEGS89] who proved that\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\gg_k n^2.\\]This question was solved in the affirmative for all $k\\geq 4$ by Bucić, Chen, and Ma [BCM26].\n\nThe situation for $C_3$ and $C_5$ are quite different - it is easy to see that\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{3})=3\\]and Erdős and Simonovits proved (as reported in [BEGS89]) that\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{5})=\\lfloor n/2\\rfloor+3\\]for all large $n$.\n\nSee also [810].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#809: [BEGS89,p.270][Er91,p.398]" }, { "number": 810, "url": "https://www.erdosproblems.com/810", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Does there exist some $\\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?", "commentary": "A problem of Burr, Erdős, Graham, and Sós [BEGS89], who believed the answer is no.\n\nEquivalently, if $\\chi_S(n,e,G)$ is the anti-Ramsey number, the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours, then this asks whether there exists an $\\epsilon>0$ such that\\[\\chi_S(n,\\epsilon n^2,C_4)\\leq n\\]for all large $n$. In [BEGS89] they prove there does not exist such $\\epsilon$ if we replace $C_4$ with $P_4$.\n\nIn [BEGS89] they ask the stronger question as to whether, for fixed $\\epsilon>0$,\\[\\chi_S(n,\\epsilon n^2,G)/n\\to \\infty\\]as $n\\to \\infty$ for all connected bipartite $G$ which is not a star. This was proved for all such $G$ except complete bipartite graphs by Sárközy and Selkow [SaSe06]. It remains open in particular for $C_4$.\n\nIn [BEGS89] they also observe that, for some constant $c>0$,\\[\\chi_S(n, cg(n;7,4),C_4)\\leq n,\\]where $g(n;7,4)$ is the maximum number of edges in a $3$-uniform hypergraph on $n$ vertices in which no $7$ vertices contain $4$ edges. It is unknown whether (but likely) that $g(n;7,4)=o(n^2)$ (see [1178]).\n\nSee also [809].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#810: [BEGS8,p.273][Er91,p.399]" }, { "number": 811, "url": "https://www.erdosproblems.com/811", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Suppose $n\\equiv 1\\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\\lfloor n/m\\rfloor$ many edges of each colours. For which graphs $G$ is it true that, if $m=e(G)$, for all large $n\\equiv 1\\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)", "commentary": "In [Er91] Erdős credits this problem to himself, Pyber, and Tuza. This problem was explored in a paper of Erdős and Tuza [ErTu93]. In [Er96] Erdős seems to suggest that this might be true for every graph $G$, and specifically asks specific challenge posed in [Er91] and [Er96] is whether, in any balanced edge-colouring of $K_{6n+1}$ by $6$ colours there must exist a rainbow $C_6$ and $K_4$.\n\nIn general, one can ask for a quantitative version, defining $d_G(n)$ to be minimal (if it exists) such that if $n$ is sufficiently large and the edges of $K_n$ are coloured with $e(G)$ many colours such that the minimum degree of each colour class is $\\geq d_G(n)$ then there is a rainbow copy of $G$. Erdős and Tuza [ErTu93] proved that\\[\\lfloor n/6\\rfloor \\leq d_{C_4}(n) \\leq \\left(\\frac{1}{4}-c\\right)n\\]for some constant $c>0$.\n\nAxenovich and Clemen [AxCl24] have proved that there exist infinitely many graphs without this property. In particular, they show that for any odd $\\ell \\geq 3$ and $m=\\lfloor \\sqrt{\\ell}+3.5\\rfloor$ there exist arbitrarily large $n$ such that $K_n$ has a balanced edge-colouring using $\\ell$ colours which contains no rainbow $K_m$. They conjecture that $K_m$ lacks this property for all $m\\geq 4$.\n\nClemen and Wagner [ClWa23] proved that $K_4$ does lack this property.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 October 2025. View history", "references": "#811: [Er91][Er93,p.346][ErTu93][Er96]" }, { "number": 812, "url": "https://www.erdosproblems.com/812", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that\\[\\frac{R(n+1)}{R(n)}\\geq 1+c\\]for some constant $c>0$, for all large $n$? Is it true that\\[R(n+1)-R(n) \\gg n^2?\\]", "commentary": "Burr, Erdős, Faudree, and Schelp [BEFS89] proved that\\[R(n+1)-R(n) \\geq 4n-8\\]for all $n\\geq 2$. The lower bound of [165] implies that\\[R(n+2)-R(n) \\gg n^{2-o(1)}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#812: [Er91]" }, { "number": 813, "url": "https://www.erdosproblems.com/813", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that\\[n^{1/3+c_1}\\ll h(n) \\ll n^{1/2-c_2}?\\]", "commentary": "A problem of Erdős and Hajnal, who could prove that\\[n^{1/3}\\ll h(n) \\ll n^{1/2}.\\]Bucić and Sudakov [BuSu23] have proved\\[h(n) \\gg n^{5/12-o(1)}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#813: [Er91]" }, { "number": 817, "url": "https://www.erdosproblems.com/817", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that\\[\\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\}\\]contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that\\[g_3(n) \\gg 3^n?\\]", "commentary": "A problem of Erdős and Sárközy who proved\\[g_3(n) \\gg \\frac{3^n}{n^{O(1)}}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#817: [Er91]" }, { "number": 819, "url": "https://www.erdosproblems.com/819", "status": "open", "prize": "no", "tags": [ "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $f(N)$ be maximal such that there exists $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lfloor N^{1/2}\\rfloor$ such that $\\lvert (A+A)\\cap [1,N]\\rvert=f(N)$. Estimate $f(N)$.", "commentary": "Erdős and Freud [ErFr91] proved\\[\\left(\\frac{3}{8}-o(1)\\right)N \\leq f(N) \\leq \\left(\\frac{1}{2}+o(1)\\right)N,\\]and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#819: [Er91]" }, { "number": 820, "url": "https://www.erdosproblems.com/820", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A263647" ], "formalized": "no", "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\\epsilon>0$,\\[H(n) > \\exp(n^{(c-\\epsilon)/\\log\\log n})\\]for infinitely many $n$ and\\[H(n) < \\exp(n^{(c+\\epsilon)/\\log\\log n})\\]for all large enough $n$?Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?", "commentary": "Erdős [Er74b] proved that there exists a constant $c>0$ such that\\[H(n) > \\exp(n^{c/(\\log\\log n)^2})\\]for infinitely many $n$.\n\nvan Doorn in the comments sketches a proof of the lower bound: that there exists some constant $c>0$ and infinitely many $n$ such that\\[H(n) > \\exp(n^{c/\\log\\log n}).\\]The sequence $H(n)$ for $1\\leq n\\leq 10$ is\\[3,3,3,6,3,18,3,6,3,12.\\]The sequence of $n$ for which $(2^n-1,3^n-1)=1$ is A263647 in the OEIS.\n\nSee also [770].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#820: [Er74b]" }, { "number": 821, "url": "https://www.erdosproblems.com/821", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A014197" ], "formalized": "no", "statement": "Let $g(n)$ count the number of $m$ such that $\\phi(m)=n$. Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that\\[g(n) > n^{1-\\epsilon}?\\]", "commentary": "Pillai proved that $\\limsup g(n)=\\infty$ and Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) >n^c$ for infinitely many $n$.\n\nThis conjecture would follow if we knew that, for every $\\epsilon>0$, there are $\\gg_\\epsilon \\frac{x}{\\log x}$ many primes $p n^{0.71568\\cdots},\\]obtained by Lichtman [Li22] as a consequence of proving that there are $\\geq \\frac{x}{(\\log x)^{O(1)}}$ many primes $p\\leq x$ such that all prime factors of $p-1$ are $\\leq x^{0.2843\\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman [BaHa98]).\n\nThe average size of $g(n)$ was investigated by Luca and Pollack [LuPo11].\n\nSee also [416].\n\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 October 2025. View history", "references": "#821: [Er74b]" }, { "number": 824, "url": "https://www.erdosproblems.com/824", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $h(x)$ count the number of integers $1\\leq ax^{2-o(1)}$?", "commentary": "Erdős [Er74b] proved that $\\limsup h(x)/x= \\infty$, and claimed a similar proof for this problem. A complete proof that $h(x)/x\\to \\infty$ was provided by Pollack and Pomerance [PoPo16].\n\nA similar question can be asked if we replace the condition $(a,b)=1$ with the condition that $a$ and $b$ are squarefree. Weisenberg suggests another variant, with the condition that there are no proper factors $u\\mid a$ and $v\\mid b$ such that $\\sigma(u)=\\sigma(v)$ and $(u,a/u)=(v,b/v)=1$, which is the weakest restriction one can impose that is still strong enough to eliminate trivial duplicates.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#824: [Er59c,p.172][Er74b,p.202]" }, { "number": 826, "url": "https://www.erdosproblems.com/826", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Are there infinitely many $n$ such that, for all $k\\geq 1$,\\[\\tau(n+k)\\ll k?\\]", "commentary": "A stronger form of [248].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#826: [Er74b]" }, { "number": 827, "url": "https://www.erdosproblems.com/827", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $n_k$ be minimal such that if $n_k$ points in $\\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\\binom{k}{3}$ triples determine circles of different radii.Determine $n_k$.", "commentary": "In [Er75h] Erdős asks whether $n_k$ exists. In [Er78c] he gave a simple argument which proves that it does, and in fact\\[n_k \\leq k+2\\binom{k-1}{2}\\binom{k-1}{3},\\]but this argument is incorrect, as explained by Martinez and Roldán-Pensado [MaRo15].\n\nMartinez and Roldán-Pensado give a corrected argument that proves $n_k\\ll k^9$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 October 2025. View history", "references": "#827: [Er75h][Er78c][Er92e]" }, { "number": 828, "url": "https://www.erdosproblems.com/828", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that, for any $a\\in\\mathbb{Z}$, there are infinitely many $n$ such that\\[\\phi(n) \\mid n+a?\\]", "commentary": "A conjecture of Graham. Lehmer has conjectured that $\\phi(n)\\mid n-1$ if and only if $n$ is prime. It is an easy exercise to show that $\\phi(n) \\mid n$ if and only if $n=2^a3^b$.\n\nThis is discussed in problem B37 of Guy's collection [Gu04]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#828: [Er83]" }, { "number": 829, "url": "https://www.erdosproblems.com/829", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subset\\mathbb{N}$ be the set of cubes. Is it true that\\[1_A\\ast 1_A(n) \\ll (\\log n)^{O(1)}?\\]", "commentary": "Mordell proved that\\[\\limsup_{n\\to \\infty} 1_A\\ast 1_A(n)=\\infty\\]and Mahler [Ma35b] proved\\[1_A\\ast 1_A(n) \\gg (\\log n)^{1/4}\\]for infinitely many $n$. Stewart [St08] improved this to\\[1_A\\ast 1_A(n) \\gg (\\log n)^{11/13}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 14 October 2025. View history", "references": "#829: [Er83]" }, { "number": 830, "url": "https://www.erdosproblems.com/830", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A259180" ], "formalized": "yes", "statement": "We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that\\[A(x)>x^{1-o(1)}?\\]", "commentary": "For example $220$ and $284$. Erdős [Er55b] proved that $A(x)=o(x)$, and Pomerance [Po81] improved this to\\[A(x) \\leq x \\exp(-(\\log x)^{1/3})\\]and later [Po15] to\\[A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2}).\\]This is problem B4 in Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#830: [Er83]" }, { "number": 831, "url": "https://www.erdosproblems.com/831", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be maximal such that in any $n$ points in $\\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.", "commentary": "See also [104] and [506].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#831: [Er75h][Er92e]" }, { "number": 835, "url": "https://www.erdosproblems.com/835", "status": "verifiable", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "yes", "statement": "Does there exist a $k>2$ such that the $k$-sized subsets of $\\{1,\\ldots,2k\\}$ can be coloured with $k+1$ colours such that for every $A\\subset \\{1,\\ldots,2k\\}$ with $\\lvert A\\rvert=k+1$ all $k+1$ colours appear among the $k$-sized subsets of $A$?", "commentary": "A problem of Erdős and Rosenfeld. This is trivially possible for $k=2$. They were not sure about $k=6$.\n\nThis is equivalent to asking whether there exists $k>2$ such that the chromatic number of the Johnson graph $J(2k,k)$ is $k+1$ (it is always at least $k+1$ and at most $2k$). The chromatic numbers listed at this website show that this is false for $3\\leq k\\leq 8$. \n\nMa and Tang have proved that the chromatic number of $J(2k,k)$ is $>k+1$ for all $k>2$ not of the form $p-1$ for prime $p$.\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#835: [Er74d,p.283]" }, { "number": 836, "url": "https://www.erdosproblems.com/836", "status": "open", "prize": "no", "tags": [ "graph theory", "hypergraphs", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).Suppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\\gg r$ many vertices?", "commentary": "A problem of Erdős and Shelah. The Fano geometry gives an example where there are no two edges which meet in $r-1$ vertices. Are there any other examples?\n\nErdős and Lovász [ErLo75] proved that there must be two edges which meet in $\\gg \\frac{r}{\\log r}$ many vertices.\n\nAlon has provided the following counterexample to the first question: as vertices take two sets $X$ and $Y$ of sizes $2r-2$ and $\\frac{1}{2}\\binom{2r-2}{r-1}$ respectively, where $Y$ corresponds to all partitions of $X$ into two equal parts. The edges are all subsets of $X$ of size $r$, and also all sets consisting of a subset of $X$ of size $r-1$ together with the unique element of $Y$ corresponding to the induced partition of $X$.\n\nThis hypergraph is intersecting, its chromatic number is $3$, and it has $\\asymp 4^r/\\sqrt{r}$ many vertices.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#836: [Er74d]" }, { "number": 837, "url": "https://www.erdosproblems.com/837", "status": "open", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with\\[\\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha\\]then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and\\[\\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta,\\]and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.What is $A_3$?", "commentary": "A problem of Erdős and Simonovits. It is known that\\[A_2 = \\left\\{ 1-\\frac{1}{k} : k\\geq 1\\right\\}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#837: [Er74d]" }, { "number": 838, "url": "https://www.erdosproblems.com/838", "status": "open", "prize": "no", "tags": [ "geometry", "convex" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be maximal such that any $n$ points in $\\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that\\[\\lim \\frac{\\log f(n)}{(\\log n)^2}=c?\\]", "commentary": "A question of Erdős and Hammer. Erdős proved in [Er78c] that there exist constants $c_1,c_2>0$ such that\\[n^{c_1\\log n}1/2$. In fact this is false - Freud [Fr93] constructed a sequence with upper density $19/36$. \n\nSee also [359] and [867].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#839: [Er78f][Er92c]" }, { "number": 840, "url": "https://www.erdosproblems.com/840", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "sidon sets" ], "oeis": [], "formalized": "no", "statement": "Let $f(N)$ be the size of the largest quasi-Sidon subset $A\\subset\\{1,\\ldots,N\\}$, where we say that $A$ is quasi-Sidon if\\[\\lvert A+A\\rvert=(1+o(1))\\binom{\\lvert A\\rvert}{2}.\\]How does $f(N)$ grow?", "commentary": "Considered by Erdős and Freud [ErFr91], who proved\\[\\left(\\frac{2}{\\sqrt{3}}+o(1)\\right)N^{1/2} \\leq f(N) \\leq \\left(2+o(1)\\right)N^{1/2}.\\](Although both bounds were already given by Erdős in [Er81h].) Note that $2/\\sqrt{3}=1.15\\cdots$. The lower bound is taking a genuine Sidon set $B\\subset [1,N/3]$ of size $\\sim N^{1/2}/\\sqrt{3}$ and taking the union with $\\{N-b : b\\in B\\}$. The upper bound was improved by Pikhurko [Pi06] to\\[f(N) \\leq \\left(\\left(\\frac{1}{4}+\\frac{1}{(\\pi+2)^2}\\right)^{-1/2}+o(1)\\right)N^{1/2}\\](the constant here is $=1.863\\cdots$).\n\nThe analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\\sim N^{1/2}$, as proved by Cilleruelo.\n\n\nSee also [30], [819], and [864].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#840: [Er81h,p.175][ErFr91][Er92c]" }, { "number": 848, "url": "https://www.erdosproblems.com/848", "status": "decidable", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is the maximum size of a set $A\\subseteq \\{1,\\ldots,N\\}$ such that $ab+1$ is never squarefree (for all $a,b\\in A$) achieved by taking those $n\\equiv 7\\pmod{25}$?", "commentary": "A problem of Erdős and Sárközy. \n\nvan Doorn has sent the following argument which shows\\[\\lvert A\\rvert \\leq (0.108\\cdots+o(1))N.\\]The condition implies, in particular, that $a^2+1$ is divisible by $p^2$ for some prime $p$ for all $a\\in A$. Furthermore, $a^2+1\\equiv 0\\pmod{p^2}$ has either $2$ or $0$ solutions, according to whether $p\\equiv 1\\pmod{4}$ or $p\\equiv 3\\pmod{4}$. It follows that every $a\\in A$ belongs to one of $2$ residue classes for some prime $p\\equiv 1\\pmod{4}$, and hence, as $N\\to \\infty$,\\[\\frac{\\lvert A\\rvert}{N}\\leq 2\\sum_{p\\equiv 1\\pmod{4}}\\frac{1}{p^2}\\approx 0.108.\\]Weisenberg has noted that this proof in fact gives the slightly better constant of $\\approx 0.105$ (see the comments section).\n\nThis was solved for all sufficiently large $N$ by Sawhney in this note. In fact, Sawhney proves something slightly stronger, that there exists some constant $c>0$ such that if $\\lvert A\\rvert \\geq (\\frac{1}{25}-c)N$ and $N$ is large then $A$ is contained in either $\\{ n\\equiv 7\\pmod{25}\\}$ or $\\{n\\equiv 18\\pmod{25}\\}$. \n\nSee also [844].\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#848: [Er92b,p.239]" }, { "number": 849, "url": "https://www.erdosproblems.com/849", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A003016", "A003015", "A059233", "A098565", "A090162", "A180058", "A182237" ], "formalized": "yes", "statement": "Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that\\[\\binom{n}{k}=a\\](with $1\\leq k\\leq n/2$) has exactly $t$ solutions?", "commentary": "Erdős [Er96b] credits this to himself and Gordon 'many years ago', but it is more commonly known as Singmaster's conjecture. For $t=3$ one could take $a=120$, and for $t=4$ one could take $a=3003$. There are no known examples for $t\\geq 5$.\n\nBoth Erdős and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions.\n\nMatomäki, Radziwill, Shao, Tao, and Teräväinen [MRSTT22] have proved that there are always at most two solutions if we restrict $k$ to\\[k\\geq \\exp((\\log n)^{2/3+\\epsilon}),\\]assuming $a$ is sufficiently large depending on $\\epsilon>0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#849: [Er96b]" }, { "number": 850, "url": "https://www.erdosproblems.com/850", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A343101" ], "formalized": "yes", "statement": "Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?", "commentary": "This is sometimes known as the Erdős-Woods conjecture.\n\nFor just $x,y$ and $x+1,y+1$ one can take\\[x=2(2^r-1)\\]and\\[y = x(x+2).\\]Erdős also asked whether there are any other examples. Makowski [Ma68] observed that $x=75$ and $y=1215$ is another example, since\\[75 = 3\\cdot 5^2 \\textrm{ and }1215 = 3^5\\cdot 5\\]while\\[76 = 2^2\\cdot 19\\textrm{ and }1216 = 2^6\\cdot 19.\\](This example was also found independently by Matthew Bolan, and by Dubickas, who posed it as part of the 2024 team selection test in Lithuania.) No other examples are known. This sequence is listed as A343101 at the OEIS.\n\nShorey and Tijdeman [ShTi16] have shown that, assuming a strong form of the ABC conjecture due to Baker, then the answer to the original problem is no.\n\nSee also [677]. \n\nThe case of $x,y$ and $x+1,y+1$ appeared as Problem 1 in the Third Benelux Mathematical Olympiad 2011. \n\nThis problem is discussed in problem B19 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#850: [Er63,Problem 60][Er80f][Er96b]" }, { "number": 852, "url": "https://www.erdosproblems.com/852", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A001223", "A053597", "A078515" ], "formalized": "no", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n(\\log x)^c\\]for some constant $c>0$, and\\[h(x)=o(\\log x)?\\]", "commentary": "Brun's sieve implies $h(x) \\to \\infty$ as $x\\to \\infty$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#852: [Er85c]" }, { "number": 853, "url": "https://www.erdosproblems.com/853", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A001223", "A390769" ], "formalized": "yes", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\\leq x$.Is it true that $r(x)\\to \\infty$? Or even $r(x)/\\log x \\to \\infty$?", "commentary": "In [Er85c] Erdős omits the condition that $t$ be even, but this is clearly necessary.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#853: [Er85c]" }, { "number": 854, "url": "https://www.erdosproblems.com/854", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A389839", "A048670" ], "formalized": "no", "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.If $1=a_1\\pi(x)+\\pi(y)+(\\log 2-o(1))\\frac{y}{(\\log y)^2},\\]where the $o(1)$ term tends to $0$ as $y\\to \\infty$.\n\nErdős [Er85c] reports Straus as remarking that the 'correct way' of stating this conjecture would have been\\[\\pi(x+y) \\leq \\pi(x)+2\\pi(y/2).\\]Clark and Jarvis [ClJa01] have shown this is also incompatible with the prime tuples conjecture.\n\nIn [Er85c] Erdős conjectures the weaker result (which in particular follows from the conjecture of Straus) that\\[\\pi(x+y) \\leq \\pi(x)+\\pi(y)+O\\left(\\frac{y}{(\\log y)^2}\\right),\\]which the Hensley and Richards result shows (conditionally) would be best possible. Richards conjectured that this is false.\n\nErdős and Richards further conjectured that the original inequality is true almost always - that is, the set of $x$ such that $\\pi(x+y)\\leq \\pi(x)+\\pi(y)$ for all $y0$.\n\nIn [Er80] Erdős conjectures that\\[\\pi(x+y)\\leq \\pi(x)+O\\left(\\frac{y}{\\log x}\\right)\\]for every $x\\geq y \\gg (\\log x)^C$ for some large constant $C>0$.\n\nHardy and Littlewood proved\\[\\pi(x+y) \\leq \\pi(x)+O(\\pi(y)).\\]The best known in this direction is a result of Montgomery and Vaughan [MoVa73], which shows\\[\\pi(x+y) \\leq \\pi(x)+2\\frac{y}{\\log y}.\\]This is discussed in problem A9 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#855: [Er61][Er65b][Er80,p.108][Er82e][Er85c]" }, { "number": 856, "url": "https://www.erdosproblems.com/856", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 3$ and $f_k(N)$ be the maximum value of $\\sum_{n\\in A}\\frac{1}{n}$, where $A$ ranges over all subsets of $\\{1,\\ldots,N\\}$ which contain no subset of size $k$ with the same pairwise least common multiple.Estimate $f_k(N)$.", "commentary": "Erdős [Er70] notes that\\[f_k(N) \\ll \\frac{\\log N}{\\log\\log N}.\\]Indeed, let $A$ be such a set. This in particular implies that, for every $t$, there are $a$. Estimate the maximum of\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}.\\]", "commentary": "Alexander [Al66] and Erdős, Sárközi, and Szemerédi [ESS68] proved that this maximum is $o(1)$ (as $N\\to \\infty$). This condition on $A$ is a weaker form of the usual primitive condition. If $A$ is primitive then Behrend [Be35] proved\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}\\ll \\frac{1}{\\sqrt{\\log\\log N}}.\\]An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.\n\nSee also [143].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#858: [Er70,p.128]" }, { "number": 859, "url": "https://www.erdosproblems.com/859", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [], "formalized": "yes", "statement": "Let $t\\geq 1$ and let $d_t$ be the density of the set of integers $n\\in\\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$.Do there exist constants $c_1,c_2>0$ such that\\[d_t \\sim \\frac{c_1}{(\\log t)^{c_2}}\\]as $t\\to \\infty$?", "commentary": "Erdős [Er70] proved that $d_t$ always exists, and that there exist some constants $c_3,c_4>0$ such that\\[\\frac{1}{(\\log t)^{c_3}} < d_t < \\frac{1}{(\\log t)^{c_4}}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#859: [Er70]" }, { "number": 860, "url": "https://www.erdosproblems.com/860", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A048670", "A058989" ], "formalized": "no", "statement": "Let $h(n)$ be such that, for any $m\\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\\leq i\\leq \\pi(n)$ such that $p_i\\mid a_i$, where $p_i$ denotes the $i$th prime. Estimate $h(n)$.", "commentary": "A problem of Erdős and Pomerance [ErPo80], who proved that\\[h(n) \\ll \\frac{n^{3/2}}{(\\log n)^{1/2}}.\\]Erdős and Selfridge proved $h(n)>(3-o(1))n$, and Ruzsa proved $h(n)/n\\to \\infty$. \n\nThis is discussed in problem B32 of Guy's collection [Gu04].\n\nSee also [375].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#860: [ErPo80][Er92c]" }, { "number": 863, "url": "https://www.erdosproblems.com/863", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)Similarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$. If $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'0$ such that, for all large $N$, if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.", "commentary": "A problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős had forgotten this). Taking all integers in $[N/8,N/4]$ and $[N/2,N]$ shows that $\\frac{5}{8}$ would be best possible here. \n\nIt is a classical folklore fact that if $A\\subseteq \\{1,\\ldots,2N\\}$ has size $\\geq N+2$ then there are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.\n\nIn general, one can define $f_k(N)$ to be minimal such that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $f_k(N)$ then there are $k$ distinct $a_i\\in A$ such that all $\\binom{k}{2}$ pairwise sums are elements of $A$. Erdős and Sós conjectured that\\[f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N,\\]and a similar example shows that this would be best possible.\n\nChoi, Erdős, and Szemerédi [CES75] have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$ such that (for large enough $N$)\\[f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#865: [Er72][CES75][Er92c]" }, { "number": 866, "url": "https://www.erdosproblems.com/866", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ such that all $\\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).Estimate $g_k(N)$.", "commentary": "A problem of Choi, Erdős, and Szemerédi. It is clear that, for the set of odd numbers in $\\{1,\\ldots,2N\\}$, no such $b_i$ exist, whence $g_k(N)\\geq 0$ always. Choi, Erdős, and Szemerédi proved that $g_3(N)=2$ and $g_4(N) \\ll 1$. van Doorn has shown that $g_4(N)\\leq 2032$.\n\nChoi, Erdős, and Szemerédi also proved that\\[g_5(N)\\asymp \\log N\\]and\\[g_6(N)\\asymp N^{1/2}.\\]In general they proved that\\[g_k(N) \\ll_k N^{1-2^{-k}}\\]and for every $\\epsilon>0$ if $k$ is sufficiently large then\\[g_k(N) > N^{1-\\epsilon}.\\]As an example, taking $A$ to be the set of all odd integers and the powers of $2$ shows that $g_5(N)\\gg \\log N$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 December 2025. View history", "references": "#866: [CES75][Er92c,p.41]" }, { "number": 869, "url": "https://www.erdosproblems.com/869", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "no", "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\\not\\in A+A$)?", "commentary": "A question of Erdős and Nathanson [ErNa88]. \n\nHärtter [Ha56] and Nathanson [Na74] proved that there exist additive bases which do not contain any minimal additive bases. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#869: [ErNa88][Er92c]" }, { "number": 870, "url": "https://www.erdosproblems.com/870", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)", "commentary": "A question of Erdős and Nathanson [ErNa79], who proved that this is true for $k=2$ if $1_A\\ast 1_A(n) > (\\log \\frac{4}{3})^{-1}\\log n$ for all large $n$.\n\nHärtter [Ha56] and Nathanson [Na74] proved that there exist additive bases which do not contain any minimal additive bases. \n\nSee also [868].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#870: [ErNa88]" }, { "number": 872, "url": "https://www.erdosproblems.com/872", "status": "open", "prize": "no", "tags": [ "number theory", "primitive sets" ], "oeis": [], "formalized": "no", "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players) such that no $a\\mid b$ for $a\\neq b\\in A$. The game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for? At least $\\epsilon n$ moves? (For $\\epsilon>0$ and $n$ sufficiently large.) At least $(1-\\epsilon)\\frac{n}{2}$ moves?", "commentary": "A number theoretic variant of a combinatorial game of Hajnal, in which players alternately add edges to a graph while keeping it triangle-free. This game must trivially end in at most $n^2/4$ moves, and Füredi and Seress [FuSe91] proved that it can be guaranteed to last for $\\gg n\\log n$ moves. Biró, Horn, and Wildstrom [BPW16] proved that it must end in at most $(\\frac{26}{121}+o(1))n^2$ moves.\n\nThis type of game is known as a saturation game.\n\nErdős does not specify which player goes first, which may result in different answers.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#872: [Er92c,p.47]" }, { "number": 873, "url": "https://www.erdosproblems.com/873", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A=\\{a_10$, there exists some $k$ such that\\[F(A,X,k)H(n)-n^{1+o(1)}?\\]Is it true that, for every $k\\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?", "commentary": "Erdős and Van Lint proved that\\[H(n)-n^{3/2-o(1)}H(n)-n^{1+o(1)}$ assuming 'plausible (but hopeless) assumptions about the distribution of primes'. They also prove the second claim when $k=2$.\n\nSee also [878].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#879: [Er84e][Er98]" }, { "number": 881, "url": "https://www.erdosproblems.com/881", "status": "open", "prize": "no", "tags": [ "number theory", "additive basis" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset\\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\\subset A$ is any infinite set then $A\\backslash B$ is not a basis of order $k$. Must there exist an infinite $B\\subset A$ such that $A\\backslash B$ is a basis of order $k+1$?", "commentary": "View the LaTeX source\n\n\n\n\n \n View history", "references": "#881: [Er98]" }, { "number": 883, "url": "https://www.erdosproblems.com/883", "status": "open", "prize": "no", "tags": [ "number theory", "graph theory" ], "oeis": [], "formalized": "no", "statement": "For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.Is it true that if\\[\\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor\\]then $G(A)$ contains all odd cycles of length $\\leq \\frac{n}{3}+1$?Is it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and\\[\\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor\\]then $G(A)$ must contain a complete $(1,\\ell,\\ell)$ triparite graph on $2\\ell+1$ vertices?", "commentary": "A problem of Erdős and Sárkőzy [ErSa97], who prove that if\\[\\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor\\]then $G(A)$ contains all odd cycles of length $\\leq cn$ for some constant $c>0$.\n\nThis threshold is the best possible, since one could take $A$ to be the set of $m\\leq n$ which are divisible by either $2$ or $3$, in which case $G(A)$ contains no triangles. \n\nThe second question was solved by Sárközy [Sa99] who proved that, for large $n$, if $\\lvert A\\rvert$ exceeds the given threshold then $G(A)$ contains a complete $(1,\\ell,\\ell)$ triparite graph with\\[\\ell \\gg \\frac{\\log n}{\\log\\log n}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 October 2025. View history", "references": "#883: [ErSa97][Er98]" }, { "number": 885, "url": "https://www.erdosproblems.com/885", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [], "formalized": "yes", "statement": "For integer $n\\geq 1$ we define the factor difference set of $n$ by\\[D(n) = \\{\\lvert a-b\\rvert : n=ab\\}.\\]Is it true that, for every $k\\geq 1$, there exist integers $N_1<\\cdots0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?", "commentary": "Erdős attributes this conjecture to Ruzsa. Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$. They also proved that, for any constant $C>0$, all large $n$ have at most $1+C^2$ many divisors in\\[[n^{1/2}, n^{1/2}+Cn^{1/4}].\\]See also [887].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#886: [ErRo97][Er98]" }, { "number": 887, "url": "https://www.erdosproblems.com/887", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [], "formalized": "yes", "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.", "commentary": "A question of Erdős and Rosenfeld [ErRo97], who proved that there are infinitely many $n$ with $4$ divisors in $(n^{1/2},n^{1/2}+n^{1/4})$, and ask whether $4$ is best possible here. They also proved that, if $n$ is large enough depending on $C$, then $n$ has at most $1+C^2$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$. \n\nChan [Ch14] has resolved this when $n$ is a square, proving that if $n$ is a square then $n$ has at most $5$ divisors in\\[[ n^{1/2}-n^{1/4}(\\log n)^{1/7}, n^{1/2}+n^{1/4}(\\log n)^{1/7}].\\]In [Ch15] Chan further proves that if $n=(N-a)(N-b)$ for some $0\\leq a\\leq b\\leq \\exp((\\log n)^{2/7})$ then $n$ has at most $18$ divisors in\\[[ n^{1/2}-n^{1/4}(\\log n)^{1/14}, n^{1/2}+n^{1/4}(\\log n)^{1/14}].\\]See also [886].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#887: [ErRo97]" }, { "number": 888, "url": "https://www.erdosproblems.com/888", "status": "open", "prize": "no", "tags": [ "number theory", "squares" ], "oeis": [ "A387584" ], "formalized": "yes", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,n\\}$ such that if $a\\leq b\\leq c\\leq d\\in A$ are such that $abcd$ is a square then $ad=bc$?", "commentary": "A question of Erdős, Sárközy, and Sós. Erdős claims that Sárközy proved that $\\lvert A\\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments).\n\nThe primes show that $\\lvert A\\rvert \\gg n/\\log n$ is possible. Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing\\[(1+o(1))\\frac{\\log\\log n}{\\log n}n \\leq \\lvert A\\rvert\\]is achievable.\n\nSee also [121].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#888: [Er98,p.178]" }, { "number": 889, "url": "https://www.erdosproblems.com/889", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "For $k\\geq 0$ and $n\\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\\leq ik$.Is it true that\\[v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty\\]as $n\\to \\infty$?", "commentary": "A question of Erdős and Selfridge [ErSe67], who could only show that $v_0(n)\\geq 2$ for $n\\geq 17$. More generally, they conjecture that\\[v_l(n)=\\max_{k\\geq l}v(n,k)\\to \\infty\\]as $n\\to \\infty$, for every fixed $l$, but could not even prove that $v_1(n)\\geq 2$ for all large $n$.\n\nThis is problem B27 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 January 2026. View history", "references": "#889: [ErSe67,p.428][Er98,p.178]" }, { "number": 890, "url": "https://www.erdosproblems.com/890", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "yes", "statement": "If $\\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\\geq 1$,\\[\\liminf_{n\\to \\infty}\\sum_{0\\leq ik$ by Pólya's theorem.\n\nIt is a classical fact that\\[\\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1.\\](There appears to be an error in [ErSe67], who ask the first question with $\\omega_k$ replaced by $\\omega$, have a bound of $k+\\pi(k)$ rather than $k$, but as explained by Meza and Tao in the comments this formulation is obviously false, and the version given here is most likely what was intended.)\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 April 2026. View history", "references": "#890: [ErSe67,p.430]" }, { "number": 891, "url": "https://www.erdosproblems.com/891", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $2=p_1k$ many prime factors?", "commentary": "Schinzel deduced from Pólya's theorem [Po18] (that the sequence of $k$-smooth integers has unbounded gaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$. \n\nThis is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?\n\nWeisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all integers at most $p_1\\cdots p_k$. By Dickson's conjecture there are infinitely many $n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m f(C) e?\\]", "commentary": "View the LaTeX source\n\n\n\n\n \n View history", "references": "#911: [Er82e,p.78]" }, { "number": 912, "url": "https://www.erdosproblems.com/912", "status": "open", "prize": "no", "tags": [ "number theory", "factorials" ], "oeis": [ "A071626" ], "formalized": "yes", "statement": "If\\[n! = \\prod_i p_i^{k_i}\\]is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. Prove that there exists some $c>0$ such that\\[h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2}\\]as $n\\to \\infty$.", "commentary": "A problem of Erdős and Selfridge, who proved (see [Er82c])\\[h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}.\\]A heuristic of Tao using the Cramér model for the primes (detailed in the comments) suggests this is true with\\[c=\\sqrt{2\\pi}=2.506\\cdots.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#912: [Er82c]" }, { "number": 913, "url": "https://www.erdosproblems.com/913", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A359747" ], "formalized": "yes", "statement": "Are there infinitely many $n$ such that if\\[n(n+1) = \\prod_i p_i^{k_i}\\]is the factorisation into distinct primes then all exponents $k_i$ are distinct?", "commentary": "It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\\{1,2,3\\}$, letting $n=8p^2-1$.\n\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#913: [Er82c,p.28]" }, { "number": 917, "url": "https://www.erdosproblems.com/917", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $k\\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).Is it true that\\[f_k(n) \\gg_k n^2?\\]Is it true that\\[f_6(n)\\sim n^2/4?\\]More generally, is it true that, for $k\\geq 6$,\\[f_k(n) \\sim \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor}\\right)n^2?\\]", "commentary": "Erdős [Er93] wrote 'I learned of this definition from Dirac in 1949 and immediately asked whether $f_k(n)=o(n^2)$. To my great surprise Dirac constructed a $6$ critical graph on $n$ vertices with more than $\\frac{n^2}{4}$ edges.' In fact Dirac [Di52] proved\\[f_6(4n+2) \\geq 4n^2+8n+3,\\]as witnessed by taking two disjoint copies of $C_{2n+1}$ and adding all edges between them.\n\nErdős [Er69b] observed that Dirac's construction generalises to show that, if $3\\mid k$, there are infinitely many values of $n$ (those of the shape $mk/3$ where $m$ is odd) such that\\[f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{k/3}\\right)n^2 + n.\\]Toft [To70] proved that $f_k(n)\\gg_k n^2$ for $k\\geq 4$.\n\nConstructions of Stiebitz [St87] show that, for $k\\geq 6$, there exist infinitely many values of $n$ such that\\[f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor+\\delta_k}\\right)n^2\\]where $\\delta_k=0$ if $k\\equiv 0\\pmod{3}$, $\\delta_k=1/7$ if $k\\equiv 1\\pmod{3}$, and $\\delta_k\\equiv 24/69$ if $k\\equiv 2\\pmod{3}$, which disproves Erdős' conjectured asympotic for $k\\not\\equiv 0\\pmod{3}$.\n\nStiebitz also proved the general upper bound\\[f_k(n) < \\mathrm{ex}(n;K_{k-1})\\sim \\frac{1}{2}\\left(1-\\frac{1}{k-2}\\right)n^2\\]for large $n$. Luo, Ma, and Yang [LMY23] have improved this upper bound to\\[f_k(n) \\leq \\frac{1}{2}\\left(1-\\frac{1}{k-2}-\\frac{1}{36(k-1)^2}+o(1)\\right)n^2\\]See also [944] and [1032].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#917: [Er69b][Er93,p.341]" }, { "number": 918, "url": "https://www.erdosproblems.com/918", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every subgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?Is there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that every subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?", "commentary": "A question of Erdős and Hajnal [ErHa68b], who proved that for every finite $k$ there is a graph with chromatic number $\\aleph_1$ where each subgraph on less than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$.\n\nIn [Er69b] it is asked with chromatic number $=\\aleph_0$, but in the comments louisd observes this is (assuming subgraph and not induced subgraph was intended) trivially impossible, and hence presumably the problem was intended as written here (which is how it is posed in [ErHa68b]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 October 2025. View history", "references": "#918: [ErHa68b][Er69b,p.28]" }, { "number": 919, "url": "https://www.erdosproblems.com/919", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Is there a graph $G$ with vertex set $\\omega_2^2$ and chromatic number $\\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\\leq \\aleph_0$?What if instead we ask for $G$ to have chromatic number $\\aleph_1$?", "commentary": "This question was inspired by a theorem of Babai, that if $G$ is a graph on a well-ordered set with chromatic number $\\geq \\aleph_0$ there is a subgraph on vertices with order-type $\\omega$ with chromatic number $\\aleph_0$.\n\nErdős and Hajnal showed this does not generalise to higher cardinals - they (see [Er69b]) constructed a set on $\\omega_1^2$ with chromatic number $\\aleph_1$ such that every strictly smaller subgraph has chromatic number $\\leq \\aleph_0$ as follows: the vertices of $G$ are the pairs $(x_\\alpha,y_\\beta)$ for $1\\leq \\alpha,\\beta <\\omega_1$, ordered lexicographically. We connect $(x_{\\alpha_1},y_{\\beta_1})$ and $(x_{\\alpha_2},y_{\\beta_2})$ if and only if $\\alpha_1<\\alpha_2$ and $\\beta_1<\\beta_2$.\n\nA similar construction produces a graph on $\\omega_2^2$ with chromatic number $\\aleph_2$ such that every smaller subgraph has chromatic number $\\leq \\aleph_1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#919: [Er69b]" }, { "number": 920, "url": "https://www.erdosproblems.com/920", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.Is it true that, for $k\\geq 4$,\\[f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}}\\]for some constant $c_k>0$?", "commentary": "Graver and Yackel [GrYa68] proved that\\[f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}.\\]It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [1104]). \n\nThe lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [166]) implies\\[f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}.\\]A positive answer to this question would follow from [986]. The known bounds for that problem imply\\[f_k(n) \\gg \\frac{n^{1-\\frac{2}{k+1}}}{(\\log n)^{c_k}}.\\]See [1104] (and also [1013]) for the case $k=3$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 21 January 2026. View history", "references": "#920: [Er69b]" }, { "number": 928, "url": "https://www.erdosproblems.com/928", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A006530" ], "formalized": "no", "statement": "Let $\\alpha,\\beta\\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n) k^{1/2-o(1)}$.\n\nIt is trivial that $S(k)\\leq k+1$ since, for example, one can take $n\\equiv 1\\pmod{(k+1)!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18] (see [4]) implies\\[S(k) \\ll k \\frac{\\log\\log\\log k}{\\log\\log k\\log\\log\\log\\log k}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 December 2025. View history", "references": "#929: [Er76d]" }, { "number": 930, "url": "https://www.erdosproblems.com/930", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Is it true that, for every $r$, there is a $k$ such that if $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then\\[\\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m\\]is not a perfect power?", "commentary": "Erdős and Selfridge [ErSe75] proved that the product of consecutive integers is never a power (establishing the case $r=1$). The condition that the intervals be large in terms of $r$ is necessary for $r=2$ - see the constructions in [363].\n\nSee also [363] for the case of squares.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#930: [Er76d]" }, { "number": 931, "url": "https://www.erdosproblems.com/931", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that\\[\\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j)\\]have the same prime factors?", "commentary": "Tijdeman gave the example\\[19,20,21,22\\textrm{ and }54,55,56,57.\\]Erdős [Er76d] was unsure of this conjecture, and thought perhaps if the two products have the same prime factors then $n_2>2(n_1+k_1)$. It is not clear but it is possible that he meant to ask this question also permitting finitely many counterexamples. Indeed, without this caveat it is false - AlphaProof has found the counterexample\\[10! = 2^8\\cdot 3^4\\cdot 5^2\\cdot 7\\]and\\[14\\cdot 15\\cdot 16 = 2^5\\cdot 3\\cdot 5\\cdot 7,\\]so that $n_1=0$, $k_1=10$, $n_2=13$, and $k_2=3$.\n\nSee also [388].\n\nThis is discussed in problem B35 of Guy's collection [Gu04]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 30 September 2025. View history", "references": "#931: [Er76d]" }, { "number": 932, "url": "https://www.erdosproblems.com/932", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A387864" ], "formalized": "yes", "statement": "Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_rn\\log n.\\]Steinerberger has noted a simple proof of this fact follows from taking $n=2^{3^r}$ for any integer $r\\geq 1$, when $k=3^r$ and $l=r+1$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#933: [Er76d]" }, { "number": 934, "url": "https://www.erdosproblems.com/934", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\\leq d$ contains two edges whose shortest path between them has length $\\geq t$.Estimate $h_t(d)$.", "commentary": "A problem of Erdős and Nešetřil. Erdős [Er88] wrote 'This problem seems to be interesting only if there is a nice expression for $h_t(d)$.'\n\nIt is easy to see that $h_t(d)\\leq 2d^t$ always and $h_1(d)=d+1$. \n\nErdős and Nešetřil and Bermond, Bond, Paoli, and Peyrat [BBPP83] independently conjectured that $h_2(d) \\leq \\tfrac{5}{4}d^2+1$, with equality for even $d$ (see [149]). This was proved by Chung, Gyárfás, Tuza, and Trotter [CGTT90].\n\nCambie, Cames van Batenburg, de Joannis de Verclos, and Kang [CCJK22] conjectured that\\[h_3(d) \\leq d^3-d^2+d+2,\\]with equality if and only if $d=p^k+1$ for some prime power $p^k$, and proved that $h_3(3)=23$. They also conjecture that, for all $t\\geq 3$, $h_t(d)\\geq (1-o(1))d^t$ for infinitely many $d$ and $h_t(d)\\leq (1+o(1))d^t$ for all $d$ (where the $o(1)$ term $\\to 0$ as $d\\to \\infty$).\n\nThe same authors prove that, if $t$ is large, then there are infinitely many $d$ such that $h_t(d) \\geq 0.629^td^t$, and that for all $t\\geq 1$ we have\\[h_t(d) \\leq \\tfrac{3}{2}d^t+1.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 October 2025. View history", "references": "#934: [Er88]" }, { "number": 935, "url": "https://www.erdosproblems.com/935", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A057521", "A389244" ], "formalized": "no", "statement": "For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\\[Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}.\\]Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then\\[Q_2(n(n+1)\\cdots(n+\\ell))2$, only keeping those prime powers with exponent $\\geq r$.\n\nThe second part of this problem is essentially the same (up to constants) as [367]. The construction given there by van Doorn (also given by Aletheia [Fe26]) proves the second part in the affirmative, since a construction via solutions to the Pell equation $x^2-8y^2=1$ implies\\[\\limsup_{n\\to \\infty}\\frac{Q_2(n(n+1)(n+2))}{n^2}=\\infty.\\]In [Fe26] it is similarly noted that the ABC conjecture implies a positive answer to the third question.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#935: [Er76d]" }, { "number": 936, "url": "https://www.erdosproblems.com/936", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A146968" ], "formalized": "yes", "statement": "Are\\[2^n\\pm 1\\]and\\[n!\\pm 1\\]powerful (i.e. if $p\\mid m$ then $p^2\\mid m$) for only finitely many $n$?", "commentary": "Cushing and Pascoe [CuPa16] have shown the answer to the second question is yes assuming the abc conjecture - in fact, for any fixed $k\\geq 0$, there are only finitely many $n$ and powerful $x$ such that $\\lvert x-n!\\rvert \\leq k$.\n\nCrowdMath [Cr20] has shown that the answer to the first question is yes, again assuming the abc conjecture.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#936: [Er76d,p.32]" }, { "number": 938, "url": "https://www.erdosproblems.com/938", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A001694" ], "formalized": "yes", "statement": "Let $A=\\{n_10$ such that\\[h(n) < (\\log n)^{c+o(1)}\\]and, for infinitely many $n$,\\[h(n) >(\\log n)^{c-o(1)}?\\]", "commentary": "Erdős writes it is not hard to prove that $\\limsup h(n)=\\infty$, and that the density $\\delta_l$ of integers for which $h(n)=l$ exists and $\\sum \\delta_l=1$. \n\nA proof that $h(n)$ is unbounded is provided by van Doorn in the comments.\n\nDe Koninck and Luca [DeLu04] have proved, for infinitely many $n$,\\[h(n) \\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/3}.\\]They also give the density ($\\approx 0.275$) of those $n$ such that $h(n)=1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 December 2025. View history", "references": "#942: [Er76d,p.35]" }, { "number": 943, "url": "https://www.erdosproblems.com/943", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $A$ be the set of powerful numbers (if $p\\mid n$ then $p^2\\mid n$). Is it true that\\[1_A\\ast 1_A(n)=n^{o(1)}\\]for every $n$?", "commentary": "View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#943: [Er76d]" }, { "number": 944, "url": "https://www.erdosproblems.com/944", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.Let $k\\geq 4$ and $r\\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?", "commentary": "A graph $G$ with chromatic number $k$ in which every vertex is critical is called $k$-vertex-critical.\n\nThis was conjectured by Dirac in 1970 for $k\\geq 4$ and $r=1$. Dirac's conjecture was proved, for $k=5$, by Brown [Br92]. Lattanzio [La02] proved there exist such graphs for all $k$ such that $k-1$ is not prime. Independently, Jensen [Je02] gave an alternative construction for all $k\\geq 5$. The case $k=4$ and $r=1$ remains open.\n\nMartinsson and Steiner [MaSt25] proved this is true for every $r\\geq 1$ if $k$ is sufficiently large, depending on $r$. Skottova and Steiner [SkSt25] have improved this, proving that such graphs exist for all $k\\geq 5$ and $r\\geq 1$. The only remaining open case is $k=4$ (even the case $k=4$ and $r=1$ remains open).\n\n\nErdős also asked a stronger quantitative form of this question: let $f_k(n)$ denote the largest $r\\geq 1$ such that there exists a $k$-vertex-critical graph on $n$ vertices such that no set of at most $r$ edges is critical. He then asks whether $f_k(n)\\to \\infty$ as $n\\to \\infty$. Skottova and Steiner [SkSt25] have proved this for $k\\geq 5$, establishing the bounds\\[n^{1/3}\\ll_k f_k(n) \\ll_k \\frac{n}{(\\log n)^C}\\]for all $k\\geq 5$, where $C>0$ is an absolute constant.\n\nThis is Problem 91 in the graph problems collection. See also [917] and [1032].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#944: [Er89e]" }, { "number": 945, "url": "https://www.erdosproblems.com/945", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [ "A048892" ], "formalized": "yes", "statement": "Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that\\[F(x) \\leq (\\log x)^{O(1)}?\\]In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?", "commentary": "A problem of Erdős and Mirsky [ErMi52], who proved that\\[\\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x) \\ll \\exp\\left(O\\left(\\frac{(\\log x)^{1/2}}{\\log\\log x}\\right)\\right).\\]Erdős [Er85e] claimed that the lower bound could be improved via their method 'with some more work' to $(\\log x)^{1-o(1)}$. Beker has improved the upper bound to\\[F(x) \\ll \\exp\\left(O\\left((\\log x)^{1/3+o(1)}\\right)\\right).\\]Cambie has observed that Cramér's conjecture implies that $F(x) \\ll (\\log x)^2$, and furthermore if every interval in $[x,2x]$ of length $\\gg \\log x$ contains a squarefree number (see [208]) then every interval of length $\\gg (\\log x)^2$ contains two numbers with the same number of divisors, whence\\[F(x) \\ll (\\log x)^2.\\]See [1004] for the analogous problem with the Euler totient function.\n\nThis problem is discussed in problem B18 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 05 October 2025. View history", "references": "#945: [ErMi52][Er85e]" }, { "number": 948, "url": "https://www.erdosproblems.com/948", "status": "open", "prize": "no", "tags": [ "number theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\\cdots$ such that $a_n0$ such that there are $\\gg n^c/\\log n$ many primes in $[n,n+n^c]$ implies that $\\liminf f(n)>0$. \n\nErdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\\epsilon>0$, if $x$ is sufficiently large there exists $y0$ then $f(n)\\ll \\log\\log\\log n$.\n\nThe study of $f(p)$ is even harder, and Erdős could not prove that\\[\\sum_{p0$.\n\nSee also [465] for a similar problem (concerning upper bounds) and [466] for a similar problem (concerning lower bounds).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#953: [Er77c]" }, { "number": 954, "url": "https://www.erdosproblems.com/954", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A390642" ], "formalized": "no", "statement": "Let $0=a_00$ such that $h(n)>n^{1+c}$ for all large $n$.", "commentary": "A problem of Erdős and Pach [ErPa90], who proved that $h(n) \\ll n^{4/3}$. They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\\ll n^{7/5}$. \n\nIt is trivial that $h(n)\\geq f(n)$, where $f(n)$ is the maximal number of unit distances determined by $n$ points in $\\mathbb{R}^2$ (see [90]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#956: [ErPa90]" }, { "number": 959, "url": "https://www.erdosproblems.com/959", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that\\[f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k).\\]Estimate\\[\\max (f(d_1)-f(d_2)),\\]where the maximum is taken over all $A$ of size $n$.", "commentary": "More generally, one can ask about\\[\\max (f(d_r)-f(d_{r+1})).\\]Clemen, Dumitrescu, and Liu [CDL25], have shown that\\[\\max (f(d_1)-f(d_2))\\gg n\\log n.\\]More generally, for any $1\\leq k\\leq \\log n$, there exists a set $A$ of $n$ points such that\\[f(d_r)-f(d_{r+1})\\gg \\frac{n\\log n}{r}.\\]They conjecture that $n\\log n$ can be improved to $n^{1+c/\\log\\log n}$ for some constant $c>0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#959: [Er84d]" }, { "number": 961, "url": "https://www.erdosproblems.com/961", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A213253" ], "formalized": "yes", "statement": "Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.", "commentary": "In other words, how large can a consecutive set of $k$-smooth integers be? Sylvester and Schur (see [Er34]) proved $f(k)\\leq k$ and Erdős [Er55d] proved\\[f(k)<3\\frac{k}{\\log k}.\\]Jutila [Ju74] and Ramachandra, and Shorey [RaSh73] proved\\[f(k) \\ll \\frac{\\log\\log\\log k}{\\log \\log k}\\frac{k}{\\log k}.\\]It is likely that $f(k) \\ll (\\log k)^{O(1)}$.\n\nThis is essentially equivalent to [683].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 April 2026. View history", "references": "#961: [Er76e,p.271]" }, { "number": 962, "url": "https://www.erdosproblems.com/962", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A327909" ], "formalized": "no", "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\\[\\log k(n) \\leq (\\log n)^{1/2+o(1)}?\\]", "commentary": "Erdős [Er65] wrote it is 'not hard to prove' that\\[\\log k(n)\\geq (\\log n)^{1/2-o(1)}\\]and it 'seems likely' that $k(n)=o(n^\\epsilon)$, but had no non-trivial upper bound for $k(n)$. In [Er76e] he gives an argument which proves\\[\\log k(n) \\gg \\sqrt{\\log n\\log\\log n},\\]and thought that this bound was 'fairly sharp'.\n\nTao in the comments has given a simple argument proving $k(n) \\leq (1+o(1))n^{1/2}$. In [Er76e] Erdős reports he could prove that\\[k(n) \\leq \\exp(-(\\log n)^c)n^{1/2}\\]for some constant $c>0$, but could not prove that $k(n) \\leq n^{1/2-c}$ for a constant $c>0$, which he said 'seems a ridiculously weak result'. \n\nTang has proved a lower bound of\\[\\log k(n)\\geq \\left(\\frac{1}{\\sqrt{2}}-o(1)\\right)\\sqrt{\\log n\\log\\log n}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 April 2026. View history", "references": "#962: [Er65][Er76e,p.273]" }, { "number": 963, "url": "https://www.erdosproblems.com/963", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be the maximal $k$ such that in any set $A\\subset \\mathbb{R}$ of size $n$ there is a subset $B\\subseteq A$ of size $\\lvert B\\rvert\\geq k$ which is dissociated that is, the sums $\\sum_{b\\in S}b$ are distinct for all $S\\subseteq B$. Estimate $f(n)$ - in particular, is it true that\\[f(n)\\geq \\lfloor \\log_2 n\\rfloor?\\]", "commentary": "Erdős noted that the greedy algorithm showed $f(n)\\geq \\lfloor \\log_3 n\\rfloor$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#963: [Er65][Va99,1.22]" }, { "number": 968, "url": "https://www.erdosproblems.com/968", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A387591" ], "formalized": "yes", "statement": "Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_nu_{n+1}$ has positive density.\n\nAs with many of Erdős' questions, by 'positive density' here he most likely meant positive lower density - in other words, there exists a constant $c>0$ such that for all large $x$ the number of such $n\\leq x$ is at least $cx$.\n\nErdős also asks whether\\[u_nu_{n+1}>u_{n+2}\\]have infinitely many solutions.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 March 2026. View history", "references": "#968: [Er65b]" }, { "number": 969, "url": "https://www.erdosproblems.com/969", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A013928" ], "formalized": "no", "statement": "Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic\\[Q(x)=\\frac{6}{\\pi^2}x+E(x).\\]", "commentary": "It is elementary to prove $E(x)\\ll x^{1/2}$, and the prime number theorem implies $o(x^{1/2})$. The best known unconditional upper bound is of the shape $x^{1/2-o(1)}$, due to Walfisz [Wa63]. Evelyn and Linfoot [EvLi31] proved that\\[E(x) \\gg x^{1/4},\\]and this is likely the true order of magnitude. The Riemann Hypothesis would follow from $E(x)\\ll x^{1/4}$. \n\nThe true order of magnitude is unknown even assuming the Riemann Hypothesis. Conditional on this assumption, the best known upper bound is\\[E(x)\\ll x^{\\frac{11}{35}+o(1)},\\]due to Liu [Li16].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#969: [Er65b][Er81h,p.176]" }, { "number": 970, "url": "https://www.erdosproblems.com/970", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A048669" ], "formalized": "no", "statement": "Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that\\[h(k) \\ll k^2?\\]", "commentary": "That $h(k)\\ll k^2$ is a conjecture of Jacobsthal. Iwaniec [Iw78] proved\\[h(k) \\ll (k\\log k)^2.\\]The best lower bound known is\\[h(k) \\gg \\frac{(\\log k)(\\log\\log\\log k)}{(\\log\\log k)^2}k,\\]due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18].\n\nThis is a more general form of the function considered in [687].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#970: [Er65b]" }, { "number": 971, "url": "https://www.erdosproblems.com/971", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A226521" ], "formalized": "yes", "statement": "Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$,\\[p(a,d) > (1+c)\\phi(d)\\log d\\]for $\\gg \\phi(d)$ many values of $a$?", "commentary": "Erdős [Er49c] could prove this is true for an infinite sequence of $d$. He also proved that, for any $\\epsilon>0$,\\[p(a,d)< \\epsilon \\phi(d)\\log d\\]for $\\gg_\\epsilon \\phi(d)$ many values of $a$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#971: [Er65b]" }, { "number": 972, "url": "https://www.erdosproblems.com/972", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $\\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\\lfloor p\\alpha\\rfloor$ is also prime?", "commentary": "Vinogradov [Vi48] proved that the sequence $\\{p\\alpha\\}$ is uniformly distributed for every irrational $\\alpha$, and hence there are infinitely many primes $p$ of the shape $p=\\lfloor n\\alpha\\rfloor$ for every irrational $\\alpha>1$. Indeed, this occurs if and only if\\[\\frac{p}{\\alpha}\\leq n<\\frac{p+1}{\\alpha},\\]which is true if and only if $\\{p\\alpha^{-1}\\}>1-\\alpha^{-1}$, which happens infinitely often by the uniform distribution of $\\{p\\alpha^{-1}\\}$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#972: [Er65b]" }, { "number": 973, "url": "https://www.erdosproblems.com/973", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence $z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with\\[\\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}?\\]", "commentary": "This is Problem 7.3 in [Ha74], where it is attributed to Erdős.\n\nErdős proved (as described on p.35 of [Tu84b]) that such a sequence does exist with $\\lvert z_i\\rvert\\leq 1$. Indeed, Erdős' construction gives a value of $C\\approx 1.32$. \n\nIn [Er92f] (a different) Erdős refines this analysis, proving that if\\[M_2=\\min_{z_j} \\max_{2\\leq k\\leq n+1} \\left\\lvert \\sum_{1\\leq j\\leq n}z_j^k\\right\\rvert,\\]where the minimum is take over all $z_j\\in \\mathbb{C}$ with $\\max \\lvert z_j\\rvert=1$, then\\[(1.746)^{-n} < M_2 < (1.745)^{-n}.\\]Tang notes in the comments that Theorem 6.1 of [Tu84b] implies that, if $\\lvert z_i\\rvert \\geq 1$ for all $i$, then\\[\\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert \\geq (2e)^{-(1+o(1))n}.\\]See also [519].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#973: [Er65b,p.213][Ha74][Va99,2.39]" }, { "number": 975, "url": "https://www.erdosproblems.com/975", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [ "A147807" ], "formalized": "yes", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\\geq 1$ for all large $n\\in\\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that\\[\\sum_{n\\leq X} \\tau(f(n))\\sim cX\\log X,\\]where $\\tau$ is the divisor function?", "commentary": "Van der Corput [Va39] proved that\\[\\sum_{n\\leq X} \\tau(f(n))\\gg_f X\\log X.\\]Erdős [Er52b] proved using elementary methods that\\[\\sum_{n\\leq X} \\tau(f(n))\\ll_f X\\log X.\\]Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley [Ho63]. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee [Mc95], [Mc97], and [Mc99] for expressions for various types of quadratic $f$). For example,\\[\\sum_{n\\leq x}\\tau(n^2+1)=\\frac{3}{\\pi}x\\log x+O(x).\\]Tao has a blog post on this topic.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#975: [Er65b]" }, { "number": 976, "url": "https://www.erdosproblems.com/976", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\\leq m\\leq n$ with $f(m)$ is divisible by a prime $\\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of\\[\\prod_{1\\leq m\\leq n}f(m).\\]Estimate $F_f(n)$. In particular, is it true that $F_f(n)\\gg n^{1+c}$ for some constant $c>0$? Or even $\\gg n^d$?", "commentary": "Nagell [Na22] and Ricci [Ri34] proved that\\[F_f(n) \\gg n\\log n,\\]which Erdős [Er52c] improved to\\[F_f(n) \\gg n(\\log n)^{\\log\\log\\log n}.\\]In [Er65b] he claimed a proof of\\[F_f(n) \\gg n\\exp((\\log n)^c)\\]for some constant $c>0$, but said he had never published the proof, which was 'fairly complicated'. This seems to have been flawed, since Erdős and Schinzel [ErSc90] later published a weaker bound. A proof of the stronger bound above was finally provided by Tenenbaum [Te90].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#976: [Er65b,p.217]" }, { "number": 978, "url": "https://www.erdosproblems.com/978", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive.Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", "commentary": "Erdős [Er53] proved there are infinitely many $n$ for which $f(n)$ is $(k-1)$-power-free, except when $k=2^l$ and $2^{k-1}\\mid f(n)$ for all $n$. This latter exception is possible, for example if\\[f(x)=k!\\left(\\binom{x}{k}+1\\right).\\]Hooley [Ho67] settled the first question, in fact providing a precise asymptotic for the number of such $n\\leq x$.\n\nHeath-Brown [He06] proved the answer to the second question is yes when $k\\geq 10$, and Browning [Br11] extended this to $k\\geq 9$ (in fact establishing an asymptotic formula for the number of such $n$). (This is under the assumption that there does not exist a prime $p$ such that $p^{k-2}\\mid f(n)$ for all $n$, which Erdős did not state explicitly, but is an obvious necessary condition, and is discussed explicitly in [He06] and [Br11].)\n\nIn [Er65b] Erdős mentions the question of whether $2^n\\pm 1$ represents infinitely many $k$th power-free integers, or $n!\\pm 1$, but that these are 'intractable at present'. (See also [936].)\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 March 2026. View history", "references": "#978: [Er65b,p.219][Er81h,p.178]" }, { "number": 979, "url": "https://www.erdosproblems.com/979", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A385316" ], "formalized": "yes", "statement": "Let $k\\geq 2$, and let $f_k(n)$ count the number of solutions to\\[n=p_1^k+\\cdots+p_k^k,\\]where the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n)=\\infty$?", "commentary": "Erdős [Er37b] proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished). \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 September 2025. View history", "references": "#979: [Er65b,p.224]" }, { "number": 982, "url": "https://www.erdosproblems.com/982", "status": "falsifiable", "prize": "no", "tags": [ "geometry", "convex", "distances" ], "oeis": [ "A004526" ], "formalized": "yes", "statement": "If $n$ distinct points in $\\mathbb{R}^2$ form a convex polygon then some vertex has at least $\\lfloor \\frac{n}{2}\\rfloor$ different distances to other vertices.", "commentary": "The regular polygon shows that $\\lfloor n/2\\rfloor$ is the best possible here.\n\nThis would be implied if there was a vertex such that no three vertices of the polygon are equally distant to it, which was originally also conjectured by Erdős [Er46b], but this is false (see [97]).\n\nLet $f(n)$ be the maximal number of such distances that are guaranteed. Moser [Mo52] proved that\\[f(n) \\geq \\left\\lceil\\frac{n}{3}\\right\\rceil.\\]This was improved by Erdős and Fishburn [ErFi94] to\\[f(n) \\geq \\left\\lfloor \\frac{n}{3}+1\\right\\rfloor,\\]then\\[f(n) \\geq \\left\\lceil \\frac{13n-6}{36}\\right\\rceil\\]by Dumitrescu [Du06b], and most recently\\[f(n) \\geq \\left(\\frac{13}{36}+\\frac{1}{22701}\\right)n-O(1)\\]by Nivasch, Pach, Pinchasi, and Zerbib [NPPZ13].\n\nIn [Er46b] Erdős makes the even stronger conjecture that on every convex curve there exists a point $p$ such that every circle with centre $p$ intersects the curve in at most $2$ points. Bárány and Roldán-Pensado [BaRo13] noted that the boundary of any acute triangle is a counterexample.\n\nBárány and Roldán-Pensado prove that, for any planar convex body, there is a point $p$ on the boundary such that every circle with centre $p$ intersects the boundary in at most $O(1)$ (where the implied constant depends on the convex body). They conjecture that there this can be bounded by an absolute constant - that is, Erdős's conjecture is true if we replace $2$ by some larger constant $C$.\n\nSee also [93].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#982: [Er46b][Er75f,p.100][Er87b,p.175][ErFi94]" }, { "number": 983, "url": "https://www.erdosproblems.com/983", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $n\\geq 2$ and $\\pi(n)r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$. Is it true that\\[2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty\\]as $n\\to \\infty$?In general, estimate $f(k,n)$, particularly when $\\pi(n)+10$ and, for any constant $1>c>0$,\\[f(cn,n)=\\log\\log n+(c_1+o(1))\\sqrt{2\\log\\log n},\\]where\\[c=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{c_1}e^{-x^2/2}\\mathrm{d}x.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 January 2026. View history", "references": "#983: [Er70b,p.138]" }, { "number": 985, "url": "https://www.erdosproblems.com/985", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A002233", "A219429", "A103309" ], "formalized": "yes", "statement": "Is it true that, for every prime $p$, there is a prime $q0$.", "commentary": "Spencer [Sp77] proved this for $k=3$ and Mattheus and Verstraete [MaVe23] proved this for $k=4$.\n\nThe best general bounds available are\\[\\frac{n^{\\frac{k+1}{2}}}{(\\log n)^{\\frac{1}{k-2}-\\frac{k+1}{2}}}\\ll_k R(k,n) \\ll_k \\frac{n^{k-1}}{(\\log n)^{k-2}}.\\]The lower bound was proved by Bohman and Keevash [BoKe10]. The upper bound was proved by Ajtai, Komlós, and Szemerédi [AKS80]. Li, Rousseau, and Zang [LRZ01] have shown that $\\ll_k$ in the upper bound can be improved to $\\leq (1+o(1))$.\n\nThe special case $k=3$ is the topic of [165] and $k=4$ is the topic of [166].\n\nThis problem is #6 in Ramsey Theory in the graphs problem collection.\n\nSee also [920].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#986: [Er90b]" }, { "number": 993, "url": "https://www.erdosproblems.com/993", "status": "falsifiable", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "The independent set sequence of any tree or forest is unimodal.In other words, if $i_k(G)$ counts the number of independent sets of vertices of size $k$ in a graph $G$, and $T$ is any tree or forest, then for some $m\\geq 0$$$i_{0}(T)\\leq i_{1}(T)\\leq\\cdots\\leq i_{m}(T)\\geq i_{m+1}(T)\\geq i_{m+2}(T)\\geq\\cdots.$$", "commentary": "A question of Alavi, Malde, Schwenk, and Erdős [AMSE87], who showed that this is false for general graphs $G$ (in fact every possible pattern of inequalities is achieved by some graph).\n\nThe sequence which counts the number of independent sets of edges of a given size was proved to be unimodal (for any graph) by Schwenk [Sc81]. In [AMSE87] they also ask whether every possible unimodal pattern of inequalities is achieved by some graph.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 February 2026. View history", "references": "#993: [AMSE87]" }, { "number": 995, "url": "https://www.erdosproblems.com/995", "status": "open", "prize": "no", "tags": [ "analysis", "discrepancy" ], "oeis": [], "formalized": "no", "statement": "Let $n_10$, for almost all $\\alpha$\\[\\limsup_{N\\to \\infty}\\frac{1}{N(\\log\\log N)^{\\frac{1}{2}-\\epsilon}}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=\\infty.\\]Erdős also proved that, for every lacunary sequence and $f\\in L^2$, for every $\\epsilon>0$, for almost all $\\alpha$,\\[\\sum_{1\\leq k\\leq N}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=o( N(\\log N)^{\\frac{1}{2}+\\epsilon}).\\]Erdős [Er64b] thought that his lower bound was closer to the truth.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#995: [Er64b]" }, { "number": 996, "url": "https://www.erdosproblems.com/996", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "Let $n_10$ such that, if\\[\\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}}\\]then\\[\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x\\]for almost every $\\alpha$?", "commentary": "Raikov proved the conclusion always holds (for every $f\\in L^2([0,1])$, with no assumption on $\\| f-f_n\\|_2$) if $n_k=a^k$ for some integer $a\\geq 2$. Erdős [Er64b] also asked whether this is true for $n_k=\\lfloor a^k\\rfloor$ for some $a>1$.\n\nKac, Salem, and Zygmund [KSZ48] proved that the conclusion holds if\\[\\| f-f_n\\|_2 \\ll \\frac{1}{(\\log n)^{c}}\\]for some constant $c>1$. Erdős [Er49d] proved that the conclusion holds if\\[\\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log n)^{c}}\\]for some constant $c>1$. Matsuyama [Ma66] improved this to $c>1/2$.\n\nIn [Er64b] Erdős asked whether the conclusion holds for all bounded functions $f$ and lacunary sequences $n_k$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#996: [Er64b]" }, { "number": 1002, "url": "https://www.erdosproblems.com/1002", "status": "open", "prize": "no", "tags": [ "analysis", "diophantine approximation" ], "oeis": [], "formalized": "no", "statement": "For any $0<\\alpha<1$, let\\[f(\\alpha,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\\{ \\alpha k\\}).\\]Does $f(\\alpha,n)$ have an asymptotic distribution function?In other words, is there a non-decreasing function $g$ such that $g(-\\infty)=0$, $g(\\infty)=1$,and\\[\\lim_{n\\to \\infty}\\lvert \\{ \\alpha\\in (0,1): f(\\alpha,n)\\leq c\\}\\rvert=g(c)?\\]", "commentary": "Kesten [Ke60] proved that if\\[f(\\alpha,\\beta,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\\{\\beta+\\alpha k\\})\\]then $f(\\alpha,\\beta,n)$ has asymptotic distribution function\\[g(c)=\\frac{1}{\\pi}\\int_{-\\infty}^{\\rho c}\\frac{1}{1+t^2}\\mathrm{d}t,\\]where $\\rho>0$ is an explicit constant.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1002: [Er64b]" }, { "number": 1003, "url": "https://www.erdosproblems.com/1003", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A001274" ], "formalized": "yes", "statement": "Are there infinitely many solutions to $\\phi(n)=\\phi(n+1)$, where $\\phi$ is the Euler totient function?", "commentary": "Erdős [Er85e] says that, presumably, for every $k\\geq 1$ the equation\\[\\phi(n)=\\phi(n+1)=\\cdots=\\phi(n+k)\\]has infinitely many solutions. \n\nErdős, Pomerance, and Sárközy [EPS87] proved that the number of $n\\leq x$ with $\\phi(n)=\\phi(n+1)$ is at most\\[\\frac{x}{\\exp((\\log x)^{1/3})}.\\]See [946] for the analogous question with the divisor function.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 October 2025. View history", "references": "#1003: [Er85e]" }, { "number": 1004, "url": "https://www.erdosproblems.com/1004", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "Let $c>0$. If $x$ is sufficiently large then does there exist $n\\leq x$ such that the values of $\\phi(n+k)$ are all distinct for $1\\leq k\\leq (\\log x)^c$, where $\\phi$ is the Euler totient function?", "commentary": "Erdős, Pomerenace, and Sárközy [EPS87] proved that if $\\phi(n+k)$ are all distinct for $1\\leq k\\leq K$ then\\[K \\leq \\frac{n}{\\exp(c(\\log n)^{1/3})}\\]for some constant $c>0$. \n\nSee [945] for the analogous problem with the divisor function.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1004: [Er85e]" }, { "number": 1005, "url": "https://www.erdosproblems.com/1005", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A386893" ], "formalized": "no", "statement": "Let $\\frac{a_1}{b_1},\\frac{a_2}{b_2},\\ldots$ be the Farey fractions of order $n\\geq 4$. Let $f(n)$ be the largest integer such that if $1\\leq k0$ such that $f(n)=(c+o(1))n$ for all large $n$?", "commentary": "The function $f(n)$ was first considered by Mayer [Ma42], who proved $f(n)\\to \\infty$ as $n\\to \\infty$. Erdős [Er43] proved $f(n)\\gg n$.\n\nvan Doorn [vD25b] has proved that\\[\\left(\\frac{1}{12}-o(1)\\right)n\\leq f(n) \\leq \\frac{1}{4}n+O(1),\\]and conjectures that the upper bound is optimal.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1005: [Er43]" }, { "number": 1011, "url": "https://www.erdosproblems.com/1011", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\\geq f_r(n)$ edges and chromatic number $\\geq r$ contains a triangle. Determine $f_r(n)$.", "commentary": "Turán's theorem implies $f_2(n)=\\lfloor n^2/4\\rfloor+1$. Erdős and Gallai [Er62d] proved $f_3(n)=\\lfloor \\frac{1}{4}(n-1)^2\\rfloor+2$.\n\nSimonovits showed in his PhD thesis (see the discussion on p. 358 of [Si74]) that\\[f_r(n)=\\frac{n^2}{4}-\\frac{g(r)}{2}{n}+O(1),\\]where $g(r)$ is the largest $m$ such that, for any triangle-free graph with chromatic number $\\geq r$, at least $m$ vertices of $G$ need to be removed to obtain a bipartite graph. Simonovits [Si74] notes\\[\\frac{\\log r}{\\log\\log r}r^2 \\ll g(r) \\ll (\\log r)^2r^2.\\]Hunter in the comments has noted that other results imply $g(r)\\asymp r^2\\log r$ - in fact\\[(1/2-o(1))r^2\\log r\\leq g(r)\\leq (2+o(1))r^2\\log r.\\]The lower bound follows from work of Davies and Illingworth [DaIl22] (see [1104]). The upper bound follows from work of Hefty, Horn, King, and Pfender [HHKP25] on $R(3,k)$.\n\nRen, Wang, Wang, and Yang [RWWY24] showed that, for $n\\geq 150$,\\[f_4(n)=\\left\\lfloor\\frac{(n-3)^2}{4}\\right\\rfloor+6.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#1011: [Er71]" }, { "number": 1013, "url": "https://www.erdosproblems.com/1013", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [ "A292528" ], "formalized": "no", "statement": "Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove\\[\\lim_{k\\to \\infty}\\frac{h_3(k+1)}{h_3(k)}=1.\\]", "commentary": "The function $h_3(k)$ is dual to the function $f(n)$ considered in [1104], in that $h_3(k)= n$ if and only if $n$ is minimal such that $f(n)=k$.\n\nGraver and Yackel [GrYa68] proved\\[h_3(k)\\gg \\frac{\\log k}{\\log\\log k}k^2.\\]The bounds for $f(n)$ from [1104] imply\\[\\left(\\frac{1}{2}-o(1)\\right)k^2\\log k\\leq h_3(k) \\leq (1+o(1))k^2\\log k.\\]See also [920] for a generalisation to $K_r$-free graphs.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 21 January 2026. View history", "references": "#1013: [Er71]" }, { "number": 1014, "url": "https://www.erdosproblems.com/1014", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices.Prove, for fixed $k\\geq 3$, that\\[\\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1.\\]", "commentary": "This is open even for $k=3$.\n\nSee also [544] for other behaviour of $R(3,k)$, and [1030] for the diagonal version of this question.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1014: [Er71,p.99]" }, { "number": 1016, "url": "https://www.erdosproblems.com/1016", "status": "open", "prize": "no", "tags": [ "graph theory", "cycles" ], "oeis": [ "A105206" ], "formalized": "no", "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $h(n)$. In particular, is it true that\\[h(n) \\geq \\log_2n+\\log_*n-O(1),\\]where $\\log_*n$ is the iterated logarithmic function?", "commentary": "Such graphs are called pancyclic. A problem of Bondy [Bo71], who claimed a proof (without details) of\\[\\log_2(n-1)-1\\leq h(n) \\leq \\log_2n+\\log_*n+O(1).\\]Erdős [Er71] believed the upper bound is closer to the truth, but could not even prove $h(n)-\\log_2n\\to \\infty$.\n\nA proof of the above lower bound is provided by Griffin [Gr13]. The first published proof of the upper bound appears to be in Chapter 4.5 of George, Khodkar, and Wallis [GKW16].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#1016: [Er71]" }, { "number": 1017, "url": "https://www.erdosproblems.com/1017", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.", "commentary": "The function $f(n,k)$ is sometimes called the clique partition number.\n\nErdős, Goodman, and Pósa [EGP66] proved that $f(n,k)\\leq n^2/4$ for all $k$ (and in fact the complete graphs can be taken to be edges and triangles), which is best possible in general, as witnessed for example by a complete bipartite graph. In [Er71] Erdő asks vaguely whether this result can be 'sharpened' for $k>n^2/4$.\n\nLovász [Lo68] proved that every graph on $n$ vertices and $k$ edges is the union of $\\binom{n}{2}-k+t$ complete graphs, where $t$ is maximal such that $t^2-t\\leq \\binom{n}{2}-k$, but without the assumption that the complete graphs are edge disjoint. Lovász's result is sharp in many cases.\n\nIf $k>n^2/4$ and the graph contains no $K_4$ then this is equivalent to finding the minimum number of edge disjoint triangles. This special case was also asked about by Erdős. A complete answer was provided by Györi and Keszegh [GyKe17], who proved that every $K_4$-free graph with $n$ vertices and $\\lfloor n^2/4\\rfloor+m$ edges contins $m$ pairwise edge disjoint triangles.\n\nSee also [184] for an analogous problem decomposing into edges and cycles, and [583] for decomposing into paths. The clique partition problem for chordal graphs is the subject of [81].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#1017: [Er71]" }, { "number": 1020, "url": "https://www.erdosproblems.com/1020", "status": "falsifiable", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges. For all $r\\geq 3$,\\[f(n;r,k)=\\max\\left(\\binom{rk-1}{r}, \\binom{n}{r}-\\binom{n-k+1}{r}\\right).\\]", "commentary": "Erdős and Gallai [ErGa59] proved this is true when $r=2$ (when $r=2$ this also follows from the Erdős-Ko-Rado theorem).\n\nThe conjectured form of $f(n;r,k)$ is the best possible, as witnessed by two examples: all $r$-edges on a set of $rk-1$ many vertices, and all edges on a set of $n$ vertices which contain at least one element of a fixed set of $k-1$ vertices. \n\nFrankl [Fr87] proved $f(n;r,k) \\leq (k-1)\\binom{n-1}{r-1}$.\n\nThis is sometimes known as the Erdős matching conjecture. Note that the second term in the maximum dominates when $n\\geq (r+1)k$. There are many partial results towards this, establishing the conjecture in different ranges. These can be separated into two regimes. For small $n$:\nThe conjecture is trivially true if $n101r^3$, and\\[kr \\leq n < k\\left(r+\\frac{1}{100r}\\right).\\]\nFor large $n$:\nErdős [Er65d] when $n>kc_r$ (where $c_r$ depends on $r$ in some unspecified fashion).\n Frankl and Füredi [Fr87] when $n>100 k^2r$.\n Bollobás, Daykin, and Erdős [BDE76] when $n\\geq 2kr^3$.\n Frankl, Rödl, and Ruciński [FRR12] when $r=3$ and $n\\geq 4k$.\n Huang, Loh, and Sudakov [HLS12] when $n\\geq 3kr^2$.\n Frankl, Luczak, and Mieczkowska [FLM12] when $n> 2k\\frac{r^2}{\\log r}$.\n Luczak and Mieczkowska [LuMi14] when $r=3$ (for all $k$).\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 December 2025. View history", "references": "#1020: [Er65d][Er71,p.103]" }, { "number": 1029, "url": "https://www.erdosproblems.com/1029", "status": "open", "prize": "$100", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then\\[\\frac{R(k)}{k2^{k/2}}\\to \\infty.\\]", "commentary": "In [Er93] Erdős offers \\$100 for a proof of this and \\$1000 for a disproof, but says 'this last offer is to some extent phoney: I am sure that [this] is true (but I have been wrong before).'\n\nErdős and Szekeres [ErSz35] proved\\[k2^{k/2} \\ll R(k) \\leq \\binom{2k-1}{k-1}.\\]One of the first applications of the probabilistic method pioneered by Erdős gives\\[R(k) \\geq (1+o(1))\\frac{1}{\\sqrt{2}e}k2^{k/2},\\]which Spencer [Sp75] improved by a factor of $2$ to\\[R(k) \\geq (1+o(1))\\frac{\\sqrt{2}}{e}k2^{k/2}.\\]See also [77] for a more general problem concerning $\\lim R(k)^{1/k}$, and discussion of upper bounds for $R(k)$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1029: [Er93,p.337]" }, { "number": 1030, "url": "https://www.erdosproblems.com/1030", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A000791", "A059442" ], "formalized": "no", "statement": "Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$.Prove the existence of some $c>0$ such that\\[\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", "commentary": "A problem of Erdős and Sós, who could not even prove whether $R(k+1,k)-R(k,k)>k^c$ for any $c>1$.\n\nIt is trivial that $R(k+1,k)-R(k,k)\\geq k-2$. Burr, Erdős, Faudree, and Schelp [BEFS89] proved\\[R(k+1,k)-R(k,k)\\geq 2k-5.\\]See also [544] for a similar question concerning $R(3,k)$, and [1014] for the general off-diagonal case.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 March 2026. View history", "references": "#1030: [Er93,p.339]" }, { "number": 1032, "url": "https://www.erdosproblems.com/1032", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.Is there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\\gg n$?", "commentary": "In [Er93] Erdős said he asked this 'more than 20 years ago'. \n\nDirac gave an example of a $6$-chromatic critical graph with minimum degree $>n/2$. This problem is also open for $5$-chromatic critical graphs.\n\nSimonovits [Si72] and Toft [To72] independently constructed $4$-chromatic critical graphs with minimum degree $\\gg n^{1/3}$. Toft conjectured that a $4$-chromatic critical graph on $n$ vertices has at least $(\\frac{5}{3}+o(1))n$ vertices, and has examples to show this would be the best possible.\n\nSee also [917] and [944].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1032: [Er93,p.341][Va99,3.60]" }, { "number": 1033, "url": "https://www.erdosproblems.com/1033", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that\\[h(n)\\geq (2(\\sqrt{3}-1)-o(1))n?\\]", "commentary": "A conjecture of Bollobás and Erdős. In [Er82e] it is asked whether $h(n)\\geq \\frac{3}{2}n$. It is now known that\\[2(\\sqrt{3}-1)n +O(1)\\geq h(n) \\geq \\frac{21}{16}n.\\]The upper bound is due to Erdős and Laskar [ErLa85], and the lower bound is due to Fan [Fa88]. (Note that $2(\\sqrt{3}-1)\\approx 1.464$ and $21/16=1.3125$.)\n\nThe upper bound is not made explicit in [ErLa85], which is concerned with chordal subgraphs (i.e. subgraphs which do not contain any induced cycle on $>3$ vertices). The connection to this problem is that taking a triangle together with all incident edges forms a chordal subgraph of $G$. The construction of the upper bound is made more explicit in [Fa88]. \n\nIn brief, let $m=\\lfloor n^2/4\\rfloor+1$ and take a complete bipartite graph between $k$ and $l$ vertices where $k+l=n$ and $k=cn$. If $c$ is chosen such that $m-kl\\leq k^2/4$ then we can add $m-kl$ edges between the $k$ vertices creating no triangles, while having maximum degree (amongst these vertices) $\\lceil 2\\frac{m-kl}{k}\\rceil$. This creates a graph $G$ on $n$ vertices with $m$ edges. Every triangle uses exactly one vertex from the side with $l$ vertices and two vertices from the side with $k$ vertices, and hence the sum of degrees is at most\\[k+2\\left\\lceil 2\\frac{m-kl}{k}\\right\\rceil+2l=(c^{-1}+3c-2)n+O(1).\\]This is minimised at $c=1/\\sqrt{3}$, where it is $2(\\sqrt{3}-1)n+O(1)$ (and a short calculation verifies that with this choice of $c$ we do indeed have $m-kl\\leq k^2/4$).\n\nMore generally, let $\\Delta_r(n,m)$ be such that every graph $G$ with $n$ vertices and $m$ edges contains a clique on $r$ vertices, say $x_1,\\ldots,x_r$, such that\\[d(x_1)+\\cdots+d(x_r)\\geq \\Delta_r(n,m).\\]Let $r\\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). One can ask more generally about the behaviour of $\\Delta_r(n,m)$ for $m>t_{r-1}(n)$ (when $m\\leq t_{r-1}(n)$ this is trivial since there may be no clique on $r$ vertices at all). The behaviour of $\\Delta_r(n,m)$ in the regime $m\\geq t_r(n)$ is the subject of [904].\n\nErdős (see [Fa92]) proved that, for every $\\epsilon>0$, there exists $\\delta>0$ such that if $t_{r-1}(n)0$, there exists $\\delta>0$ such that if $m>t_r(n)-\\delta n^2$ then\\[\\Delta_r(n,m)\\geq (1-\\epsilon)\\frac{2rm}{n}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 April 2026. View history", "references": "#1033: [Er82e,p.71][Er93,p.344]" }, { "number": 1035, "url": "https://www.erdosproblems.com/1035", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?", "commentary": "Erdős [Er93] says 'if the conjecture is false, two related problems could be asked':\nDetermine or estimate the smallest $m>2^n$ such that every graph on $m$ vertices with minimum degree $>(1-c)2^n$ contains a $Q_n$, and \nFor which $u_n$ is it true that every graph on $2^n$ vertices with minimum degree $>2^n-u_n$ contains a $Q_n$.\nSee also [576] for the extremal number of edges that guarantee a $Q_n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1035: [Er93,p.345]" }, { "number": 1038, "url": "https://www.erdosproblems.com/1038", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "yes", "statement": "Determine the infimum and supremum of\\[\\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\]as $f\\in \\mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.", "commentary": "A problem of Erdős, Herzog, and Piranian [EHP58], who proved that the measure of the set in question is always at most $2\\sqrt{2}$ under the assumption that all the roots are in $\\{-1,1\\}$, and conjecture this is the best possible upper bound.\n\nThey also note that the infimum of the set in question is less than $2$, as witnessed by $f(x)=(x+1)(x-1)^m$ for $m\\geq 3$. They further note that if the roots are restricted to $[-2,2]$ then the infimum is zero, as witnessed by a small perturbation of the Chebyshev polynomials. \n\nThey further conjectured that, if the roots are restricted to $[-2,2]$, then\\[\\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\geq n^{-c}\\]for an absolute constant $c>0$. This was proved by Pommerenke [Po61], who in fact showed that this set must contain an interval of width $\\gg n^{-4}$.\n\nThe current best known bounds (see the discussion in the comments) are\\[1.519\\approx 2^{4/3}-1\\leq \\inf \\leq 1.835\\cdots\\]and\\[\\sup = 2\\sqrt{2}\\approx 2.828.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 January 2026. View history", "references": "#1038: [EHP58,p.131]" }, { "number": 1039, "url": "https://www.erdosproblems.com/1039", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert \\leq 1$ for all $i$. Let $\\rho(f)$ be the radius of the largest disc which is contained in $\\{z: \\lvert f(z)\\rvert< 1\\}$. Determine the behaviour of $\\rho(f)$. In particular, is it always true that $\\rho(f)\\gg 1/n$?", "commentary": "A problem of Erdős, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\\rho(f) \\leq \\frac{\\pi/2}{n}$.\n\nPommerenke [Po61] proved that\\[\\rho(f) \\geq \\frac{1}{2en^2}.\\]Krishnapur, Lundberg, and Ramachandran [KLR25] proved\\[\\rho(f) \\gg \\frac{1}{n\\sqrt{\\log n}}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#1039: [EHP58,p.134]" }, { "number": 1040, "url": "https://www.erdosproblems.com/1040", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Let $F\\subseteq \\mathbb{C}$ be a closed infinite set, and let $\\mu(F)$ be the infimum of\\[\\lvert \\{ z: \\lvert f(z)\\rvert < 1\\}\\rvert,\\]as $f$ ranges over all polynomials of the shape $\\prod (z-z_i)$ with $z_i\\in F$.Is $\\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\\mu(F)=0$ whenever the transfinite diameter of $F$ is $\\geq 1$?", "commentary": "A problem of Erdős, Herzog, and Piranian [EHP58], who show that the answer is yes if $F$ is a line segment or disc, and that if the transfinite diameter is $<1$ then $\\{ z: \\lvert f(z)\\rvert < 1\\}$ always contains a disc of radius $\\gg_F 1$.\n\nErdős and Netanyahu [ErNe73] proved that if $F$ is also bounded and connected, with transfinite diameter $00$. The best value of $C$ available is $C\\approx 1.268$, proved in [CDDHT26]. If we restrict to $n$ divisible by $6$ this can be improved to $C\\approx 1.304$.\n\nIt remains possible that the regular polygon is a maximiser for odd $n$; this is explicitly conjectured in [CDDHT26]. It is not yet known whether, for odd $n$,\\[\\lim \\frac{\\max \\Delta}{n^n}=e^{\\pi^2/8}\\approx 3.433.\\]The paper [CDDHT26] also contains a number of structural results about the form of an extremal set of points.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 April 2026. View history", "references": "#1045: [EHP58,p.143]" }, { "number": 1049, "url": "https://www.erdosproblems.com/1049", "status": "open", "prize": "no", "tags": [ "irrationality" ], "oeis": [], "formalized": "yes", "statement": "Let $t>1$ be a rational number. Is\\[\\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}\\]irrational, where $\\tau(n)$ counts the divisors of $n$?", "commentary": "A conjecture of Chowla. Erdős [Er48] proved that this is true if $t\\geq 2$ is an integer.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#1049: [Er88c,p.102]" }, { "number": 1052, "url": "https://www.erdosproblems.com/1052", "status": "open", "prize": "$10", "tags": [ "number theory" ], "oeis": [ "A002827" ], "formalized": "yes", "statement": "A unitary divisor of $n$ is $d\\mid n$ such that $(d,n/d)=1$. A number $n\\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (aside from $n$ itself).Are there only finite many unitary perfect numbers?", "commentary": "Guy [Gu04] reports that Carlitz, Erdős, and Subbarao offer \\$10 for settling this question, and that Subbarao offers 10 cents for each new example.\n\nThere are no odd unitary perfect numbers. There are five known unitary perfect numbers (A002827 in the OEIS):\\[6, 60, 90, 87360, 146361946186458562560000.\\]This is problem B3 in Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#1052: [Gu04]" }, { "number": 1053, "url": "https://www.erdosproblems.com/1053", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A007539" ], "formalized": "no", "statement": "Call a number $k$-perfect if $\\sigma(n)=kn$, where $\\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\\log\\log n)$?", "commentary": "A question of Erdős, as reported in problem B2 of Guy's collection [Gu04]. Guy further writes 'It has even been suggested that there may be only finitely many $k$-perfect numbers with $k\\geq 3$.' The largest $k$ for which a $k$-perfect number has been found is $k=11$ - see this page for more information.\n\nThese are known as multiply perfect numbers. When $k=2$ this is the definition of a perfect number.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1053: [Gu04]" }, { "number": 1054, "url": "https://www.erdosproblems.com/1054", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [ "A167485" ], "formalized": "yes", "statement": "Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\\geq 1$.Is it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\\limsup f(n)/n=\\infty$?", "commentary": "A question of Erdős reported in problem B2 of Guy's collection [Gu04]. The function $f(n)$ is undefined for $n=2$ and $n=5$, but is likely well-defined for all $n\\geq 6$ (which would follow from a strong form of Goldbach's conjecture).\n\nThe sequence of values of $f(n)$ is given by A167485 in the OEIS.\n\nSee also [468].\n\nThe strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to [468], in which he proves that the upper density of $\\{ n : f(n)\\leq \\delta n\\}$ is $\\ll \\delta^2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#1054: [Gu04]" }, { "number": 1055, "url": "https://www.erdosproblems.com/1055", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A005113" ], "formalized": "yes", "statement": "A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.Are there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?", "commentary": "A classification due to Erdős and Selfridge. It is easy to prove that the number of primes $\\leq n$ in class $r$ is at most $n^{o(1)}$. \n\nThe sequence $p_r$ begins $2,13,37,73,1021$ (A005113 in the OEIS). Erdős thought $p_r^{1/r}\\to \\infty$, while Selfridge thought it quite likely to be bounded.\n\nA similar question can be asked replacing $p+1$ with $p-1$.\n\nThis is problem A18 in Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1055: [Er77]" }, { "number": 1056, "url": "https://www.erdosproblems.com/1056", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A060427" ], "formalized": "yes", "statement": "Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that\\[\\prod_{n\\in I_i}n \\equiv 1\\pmod{p}\\]for all $1\\leq i\\leq k$?", "commentary": "This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that\\[3\\cdot 4\\equiv 5\\cdot 6\\cdot 7\\equiv 1\\pmod{11},\\]establishing the case $k=2$. Makowski [Ma83] found, for $k=3$,\\[2\\cdot 3\\cdot 4\\cdot 5\\equiv 6\\cdot 7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\equiv 12\\cdot 13\\cdot 14\\cdot 15\\equiv 1\\pmod{17}.\\]Noll and Simmons asked, more generally, whether there are solutions to $q_1!\\equiv\\cdots \\equiv q_k!\\pmod{p}$ for arbitrarily large $k$ (with $q_1<\\cdots0$. Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact\\[C(x)= x \\exp\\left(-(1+o(1))\\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right).\\]Alford, Granville, and Pomerance [AGP94] proved that $C(x)\\to \\infty$, and in fact $C(x)>x^{2/7}$ for large $x$. The lower bound\\[C(x)> x^{0.33336704}\\]was proved by Harman [Ha08]. This exponent was improved to $0.3389$ by Lichtman [Li22].\n\nKorselt observed that $n$ being a Carmichael number is equivalent to $n$ being squarefree and $p-1\\mid n-1$ for all primes $p\\mid n$.\n\nThis is discussed in problem A13 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 January 2026. View history", "references": "#1057: [Er56c]" }, { "number": 1059, "url": "https://www.erdosproblems.com/1059", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A064152" ], "formalized": "yes", "statement": "Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\\leq k!l$, and all the numbers $n-k!$ are composite for $1\\leq k\\leq l$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1059: [Gu04]" }, { "number": 1060, "url": "https://www.erdosproblems.com/1060", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A327153" ], "formalized": "yes", "statement": "Let $f(n)$ count the number of solutions to $k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\\leq n^{o(\\frac{1}{\\log\\log n})}$? Perhaps even $\\leq (\\log n)^{O(1)}$?", "commentary": "This is discussed in problem B11 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 28 September 2025. View history", "references": "#1060: [Gu04]" }, { "number": 1061, "url": "https://www.erdosproblems.com/1061", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A110177" ], "formalized": "yes", "statement": "How many solutions are there to\\[\\sigma(a)+\\sigma(b)=\\sigma(a+b)\\]with $a+b\\leq x$, where $\\sigma$ is the sum of divisors function? Is it $\\sim cx$ for some constant $c>0$?", "commentary": "A question of Erdős reported in problem B15 of Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1061: [Gu04]" }, { "number": 1062, "url": "https://www.erdosproblems.com/1062", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A038372" ], "formalized": "yes", "statement": "Let $f(n)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,n\\}$ such that there are no three distinct elements $a,b,c\\in A$ such that $a\\mid b$ and $a\\mid c$. How large can $f(n)$ be? Is $\\lim f(n)/n$ irrational?", "commentary": "The example $[m+1,3m+2]$ shows that $f(n)\\geq\\lceil \\frac{2}{3}n\\rceil$. Lebensold [Le76] has shown that, for large $n$,\\[0.6725 n \\leq f(n) \\leq 0.6736 n.\\]This is problem B24 in Guy's collection [Gu04].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 January 2026. View history", "references": "#1062: [Gu04]" }, { "number": 1063, "url": "https://www.erdosproblems.com/1063", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A389360" ], "formalized": "yes", "statement": "Let $k\\geq 2$ and define $n_k\\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\\binom{n}{k}$ for all but one $0\\leq i1$. Is it true that $g_d(n) \\gg n/d$ in general? The upper bound $g_d(n) \\ll n/d$ is trivial, considering widely spaced unit simplices.\n\nSee [1070] for the general estimate of independence number of unit distance graphs.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 02 October 2025. View history", "references": "#1066: [Er87b,p.171]" }, { "number": 1068, "url": "https://www.erdosproblems.com/1068", "status": "open", "prize": "no", "tags": [ "graph theory", "set theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?", "commentary": "This is described in [BoPi24] as a 'version of the Erdős-Hajnal problem' (which is [1067]), but it does not seem to appear in [ErHa66].\n\nWe say a graph is infinitely (vertex) connected if any two vertices are connected by infinitely many pairwise vertex-disjoint paths.\n\nSoukup [So15] constructed a graph with uncountable chromatic number in which every uncountable set is finitely vertex-connected. A simpler construction was given by Bowler and Pitz [BoPi24].\n\nSee also [1067].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1068: [Va99,7.90]" }, { "number": 1070, "url": "https://www.erdosproblems.com/1070", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be maximal such that, given any $n$ points in $\\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\\geq n/4$?", "commentary": "In other words, estimate the minimal independence number of a unit distance graph with $n$ vertices. If $\\omega$ is the independence number and $\\chi$ is the chromatic number then $\\omega \\chi\\geq n$, and hence $f(n)\\geq n/\\chi$, where $\\chi$ is the answer to the Hadwiger-Nelson problem [508].\n\nThe Moser spindle shows $f(n)\\leq \\frac{2}{7}n\\approx 0.285n$. Larman and Rogers [LaRo72] noted that if $m_1$ is the supremum of the upper densities of measurable subsets of $\\mathbb{R}^2$ which have no unit distance pairs then\\[f(n)\\geq m_1n.\\]Croft [Cr67] gave the best-known lower bound of $m_1\\geq 0.22936$ and hence\\[0.22936n \\leq f(n) \\leq \\frac{2}{7}n\\approx 0.285n.\\]Ambrus, Csiszárik, Matolcsi, Varga, and Zsámboki [ACMVZ23] have proved that $m_1\\leq 0.247$, and hence this approach cannot achieve $f(n)\\geq n/4$. See [232] for more on $m_1$. \n\nMatolcsi, Ruzsa, Varga, and Zsámboki [MRVZ23] have improved the upper bound to\\[f(n) \\leq \\left(\\frac{1}{4}+o(1)\\right)n.\\]They conjecture that $m_1=0.22936\\cdots$ (the lower bound of Croft mentioned above) and $f(n)=(1/4+o(1))n$.\n\nIf we also insist that no two points are distance $<1$ apart then this is problem becomes [1066].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 22 January 2026. View history", "references": "#1070: [Er87b,p.171]" }, { "number": 1072, "url": "https://www.erdosproblems.com/1072", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A073944", "A072937", "A154554" ], "formalized": "yes", "statement": "For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\\equiv 0\\pmod{p}$.Is it true that there are infinitely many $p$ for which $f(p)=p-1$?Is it true that $f(p)/p\\to 0$ for almost all $p$?", "commentary": "Questions formulated by Erdős, Hardy, and Subbarao [HaSu02], who believed that the number of $p\\leq x$ for which $f(p)=p-1$ is $o(x/\\log x)$. \n\nThese are mentioned in problem A2 of Guy's collection.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 October 2025. View history", "references": "#1072: [HaSu02][Gu04]" }, { "number": 1073, "url": "https://www.erdosproblems.com/1073", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A256519" ], "formalized": "yes", "statement": "Let $A(x)$ count the number of composite $ur^{-r}$ such that, for any $\\epsilon>0$, if $n$ is sufficiently large, the following holds.Any $r$-uniform hypergraph on $n$ vertices with at least $(1+\\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\\to \\infty$ as $n\\to \\infty$.", "commentary": "Erdős [Er64f] proved that this is true with $c_r=r^{-r}$ whenever the graph has at least $\\epsilon n^r$ many edges.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 05 October 2025. View history", "references": "#1075: [Er74c,p.80]" }, { "number": 1082, "url": "https://www.erdosproblems.com/1082", "status": "falsifiable", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\\lfloor n/2\\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\\lfloor n/2\\rfloor$ distinct distances?", "commentary": "A conjecture of Szemerédi, who proved this with $n/2$ replaced by $n/3$. More generally, Szemerédi gave a simple proof that if there are no $k$ points on a line then some point determines $\\gg n/k$ distinct distances (a weak inverse result to the distinct distance problem [89]).\n\nThis is a stronger form of [93]. The second question is a stronger form of [982].\n\nSzemerédi's proof is unpublished, but given in [Er75f].\n\nIn [Er75f] Erdős asks whether, given $n$ points in $\\mathbb{R}^3$ with no three on a line, do they determine $\\gg n$ distances? Altman proved the answer is yes if the points form the vertices of a convex polyhedron (see [660] for a stronger form of this), and Szemerédi proved the answer is yes if there are no four points on a plane.\n\nThe second stronger question has a negative answer: there is a construction of $8$ points such that each point has exactly three distinct distances to the others. This first appeared in the literature in a paper of Erdős and Fishburn [ErFi97b], where they credit the construction to Harborth. This configuration is studied in detail by Fishburn [Fi02].\n\n(This construction was later found by DeepMind. An earlier construction was also given by Xichuan in the comments, who gave a set of $42$ points in $\\mathbb{R}^2$, with no three on a line, such that each point determines only $20$ distinct distances.)\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#1082: [Er75f,p.101][Er87b][Er97e]" }, { "number": 1083, "url": "https://www.erdosproblems.com/1083", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that\\[f_d(n)=n^{\\frac{2}{d}-o(1)}?\\]", "commentary": "A generalisation of the distinct distance problem [89] to higher dimensions. Erdős [Er46b] proved\\[n^{1/d}\\ll_d f_d(n)\\ll_d n^{2/d},\\]the upper bound construction being given by a set of lattice points.\n Clarkson, Edelsbrunner, Gubias, Sharir, and Welzl [CEGSW90] proved $f_3(n)\\gg n^{1/2}$.\nAronov, Pach, Sharir, and Tardos [APST04] proved $f_d(n)\\gg n^{\\frac{1}{d-90/77}-o(1)}$ for any $d\\geq 3$ (for example, $f_3(n)\\gg n^{0.546}$).\nSolymosi and Vu [SoVu08] proved $f_3(n) \\gg n^{3/5}$ and\\[ f_d(n)\\gg_d n^{\\frac{2}{d}-\\frac{c}{d^2}}\\]for all $d\\geq 4$ for some constant $c>0$. (The result in their paper for $d=3$ is slightly weaker than stated here, but uses as a black box the bound for distinct distances in $2$ dimensions; we have recorded the consequence of combining their method with the work of Guth and Katz on [89].)\nThe function $f_d(n)$ is essentially the inverse of the function $g_d(n)$ considered in [1089] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)0$, which the triangular lattice shows is the best possible up to the value of $c$. In [Er75f] he speculated that the triangular lattice is exactly the best possible, and in particular\\[f_2(3n^2+3n+1)=9n^2+3n.\\]Harborth [Ha74b] proved this, and more generally\\[f_2(n)=\\lfloor 3n-\\sqrt{12n-3}\\rfloor\\]for all $n\\geq 2$.\n\nIn [Er75f] he claims the existence of $c_1,c_2>0$ such that\\[6n-c_1n^{2/3}< f_3(n) < 6n-c_2n^{2/3}.\\]An upper bound of\\[f_3(n) < 6n-0.926n^{2/3}\\]for all $n\\geq 2$ was proved by Bezdek and Reid [BeRe13].\n\nIn general, it is known that\\[(d-o(1))n \\leq f_d(n) \\leq 2^{O(d)}n,\\]the lower bound coming from points arranged in an integer grid and the upper bound from the fact that $2^{O(d)}$ many non-intersecting congruent balls can touch a fixed ball (the kissing number problem).\n\nA recent survey on contact numbers for sphere packings is by Bezdek and Khan [BeKh18].\n\nSee [223] for the analogous problem with maximal distance $1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 February 2026. View history", "references": "#1084: [Er75f,p.102]" }, { "number": 1085, "url": "https://www.erdosproblems.com/1085", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "yes", "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.", "commentary": "The most difficult cases are $d=2$ and $d=3$. When $d=2$ this is the unit distance problem [90], and the best known bounds are\\[n^{1+\\frac{c}{\\log\\log n}}< f_2(n) \\ll n^{4/3}\\]for some constant $c>0$, the lower bound by Erdős [Er46b] and the upper bound by Spencer, Szemerédi, and Trotter [SST84].\n\nWhen $d=3$ the best known bounds are\\[n^{4/3}\\log\\log n \\ll f_3(n) \\ll n^{3/2}\\beta(n)\\]where $\\beta(n)$ is a very slowly growing function, the lower bound by Erdős [Er60b] and the upper bound by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl [CEGSW90].\n\nA construction of Lenz (taking points on orthogonal circles) shows that, for $d\\geq 4$,\\[f_d(n)\\geq \\frac{p-1}{2p}n^2-O(1)\\]with $p=\\lfloor d/2\\rfloor$. Erdős [Er60b] showed that the Erdős-Stone theorem implies\\[f_d(n) \\leq \\left(\\frac{p-1}{2p}+o(1)\\right)n^2\\]for $d\\geq 4$.\n\nErdős [Er67e] determined $f_d(n)$ up to $O(1)$ for all even $d\\geq 4$. Brass [Br97] determined $f_4(n)$ exactly. Swanepoel [Sw09] determined $f_d(n)$ exactly for even $d\\geq 6$. For odd $d\\geq 5$ Erdős and Pach [ErPa90] proved that there exist constants $c_1(d),c_2(d)>0$ such that\\[\\frac{p-1}{2p}n^2 +c_1n^{4/3}\\leq f_d(n) \\leq \\frac{p-1}{2p}n^2 +c_2n^{4/3}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 17 October 2025. View history", "references": "#1085: [Er75f,p.103]" }, { "number": 1086, "url": "https://www.erdosproblems.com/1086", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $g(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.", "commentary": "Equivalently, how many triangles of area $1$ can a set of $n$ points in $\\mathbb{R}^2$ determine? Erdős and Purdy attribute this question to Oppenheim. Erdős and Purdy [ErPu71] proved\\[n^2\\log\\log n \\ll g(n) \\ll n^{5/2},\\]and believed the lower bound to be closer to the truth. The upper bound has been improved a number of times - by Pach and Sharir [PaSh92], Dumitrescu, Sharir, and Tóth [DST09], Apfelbaum and Sharir [ApSh10], and Apfaulbaum [Ap13]. The best known bound is\\[g(n) \\ll n^{20/9}\\]by Raz and Sharir [RaSh17].\n\nErdős and Purdy also ask a similar question about the higher-dimensional generalisations - more generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ many $r$-dimensional simplices with the same volume.\n\nErdős and Purdy [ErPu71] proved $g_3^2(n) \\ll n^{8/3}$, and Dumitrescu, Sharir, and Tóth [DST09] improved this to $g_3^2(n) \\ll n^{2.4286}$. \nErdős and Purdy [ErPu71] proved $g_6^2(n)\\gg n^3$. Purdy [Pu74] proved\\[g_4^2(n)\\leq g^2_5(n) \\ll n^{3-c}\\]for some constant $c>0$. An observation of Oppenheim (using a construction of Lenz) detailed in [ErPu71] shows that\\[g_{2k+2}^k(n)\\geq \\left(\\frac{1}{(k+1)^{k+1}}+o(1)\\right)n^{k+1}\\]and Erdős and Purdy conjecture this is the best possible.\n\nSee also [90] and [755].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 October 2025. View history", "references": "#1086: [ErPu71][Er75f,p.104]" }, { "number": 1087, "url": "https://www.erdosproblems.com/1087", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\\leq n^{3+o(1)}$?", "commentary": "A question of Erdős and Purdy [ErPu71], who proved\\[n^3\\log n \\ll f(n) \\ll n^{7/2}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1087: [ErPu71][Er75f,p.104]" }, { "number": 1088, "url": "https://www.erdosproblems.com/1088", "status": "open", "prize": "no", "tags": [ "geometry" ], "oeis": [], "formalized": "no", "statement": "Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\\geq 3$,\\[f_d(n)=2^{o(d)}?\\]", "commentary": "It is easy to prove that $f_d(n) \\leq n^{O_d(1)}$. Erdős [Er75f] claimed that he and Straus proved $f_d(n)\\leq c_n^d$ for some constant $c_n>0$.\n\nWhen $d=1$ this is the subject of [530], and $f_1(n)\\asymp n^2$.\n\nWhen $n=3$ this is the subject of [503]. Erdős could prove $f_2(3)=7$ and Croft [Cr62] proved $f_3(3)=9$. The results described at [503] demonstrate that $f_d(3)=d^2/2+O(d)$. \n\nThe behaviour of $f_d(n)$ for fixed $d$ as $n\\to \\infty$ is the subject of [1208].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1088: [Er75f,p.104]" }, { "number": 1092, "url": "https://www.erdosproblems.com/1092", "status": "open", "prize": "no", "tags": [ "geometry", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$.Is it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", "commentary": "A conjecture of Erdős, Hajnal, and Szemerédi. This seems to be closely related to, but distinct from, [744].\n\nTang notes in the comments that a construction of Rödl [Ro82] disproves the first question, so that $f_2(n)\\not\\gg n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 December 2025. View history", "references": "#1092: [Er76c,p.4]" }, { "number": 1093, "url": "https://www.erdosproblems.com/1093", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [], "formalized": "yes", "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", "commentary": "A problem of Erdős, Lacampagne, and Selfridge [ELS88], that was also asked in the 1986 problem session of West Coast Number Theory (as reported here).\n\nIn [ELS93] they prove that if the deficiency exists and is $\\geq 1$ then $n\\ll 2^k\\sqrt{k}$. \n\nThe following examples are either from [ELS88] or here. The following have deficiency $1$ (there are $58$ examples with $n\\leq 10^5$):\\[\\binom{7}{3},\\binom{13}{4},\\binom{14}{4},\\binom{23}{5},\\binom{62}{6},\\binom{94}{10},\\binom{95}{10}.\\]The examples which follow are the only known examples with deficiency $>1$. The following have deficiency $2$:\\[\\binom{44}{8},\\binom{74}{10},\\binom{174}{12},\\binom{239}{14},\\binom{5179}{27},\\binom{8413}{28},\\binom{8414}{28},\\binom{96622}{42}.\\]The following have deficiency $3$:\\[\\binom{46}{10},\\binom{47}{10},\\binom{241}{16},\\binom{2105}{25},\\binom{1119}{27},\\binom{6459}{33}.\\]The following has deficiency $4$:\\[\\binom{47}{11}.\\]The following has deficiency $9$:\\[\\binom{284}{28}.\\]See also [384] and [1094].\n\nBarreto in the comments has given a positive answer to the second question, conditional on two (strong) conjectures.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 December 2025. View history", "references": "#1093: [ELS88,p.522]" }, { "number": 1094, "url": "https://www.erdosproblems.com/1094", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [], "formalized": "yes", "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "commentary": "A stronger form of [384] that appears in a paper of Erdős, Lacampagne, and Selfridge [ELS88]. Erdős observed that the least prime factor is always $\\leq n/k$ provided $n$ is sufficiently large depending on $k$. Selfridge [Se77] further conjectured that this always happens if $n\\geq k^2-1$, except $\\binom{62}{6}$.\n\nThe threshold $g(k)$ below which $\\binom{n}{k}$ is guaranteed to be divisible by a prime $\\leq k$ is the subject of [1095].\n\nMore precisely, in [ELS88] they conjecture that if $n\\geq 2k$ then the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$ with the following $14$ exceptions:\\[\\binom{7}{3},\\binom{13}{4},\\binom{23}{5},\\binom{14}{4},\\binom{44}{8},\\binom{46}{10},\\binom{47}{10},\\]\\[\\binom{47}{11},\\binom{62}{6},\\binom{74}{10},\\binom{94}{10},\\binom{95}{10},\\binom{241}{16},\\binom{284}{28}.\\]They also suggest the stronger conjecture that, with a finite number of exceptions, the least prime factor is $\\leq \\max(n/k,\\sqrt{k})$, or perhaps even $\\leq \\max(n/k,O(\\log k))$. Indeed, in [ELS93] they provide some further computational evidence, and point out it is consistent with what they know that in fact this holds with $\\leq \\max(n/k,13)$, with only $12$ exceptions.\n\nDiscussed in problem B31 and B33 of Guy's collection [Gu04] - there Guy credits Selfridge with the conjecture that if $n> 17.125k$ then $\\binom{n}{k}$ has a prime factor $p\\leq n/k$. \n\nThis is related to [1093], in that the only counterexamples to this conjecture can occur from $\\binom{n}{k}$ with deficiency $\\geq 1$.\n\nThere is an interesting discussion about this problem on MathOverflow.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 October 2025. View history", "references": "#1094: [ELS88][ELS93]" }, { "number": 1095, "url": "https://www.erdosproblems.com/1095", "status": "open", "prize": "no", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A003458" ], "formalized": "yes", "statement": "Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$. Estimate $g(k)$.", "commentary": "A question of Ecklund, Erdős, and Selfridge [EES74], who proved\\[k^{1+c}0$, and conjectured $g(k)0$, due to Konyagin [Ko99b].\n\nErdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.\n\nSorenson, Sorenson, and Webster [SSW20] give heuristic evidence that\\[\\log g(k) \\asymp \\frac{k}{\\log k}.\\]See also [1094].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 12 January 2026. View history", "references": "#1095: [EES74]" }, { "number": 1096, "url": "https://www.erdosproblems.com/1096", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", "commentary": "A problem of Erdős and Joó posed in the 1991 problem session of Great Western Number Theory.\n\nThey speculate that the threshold may be $q_0$, where $q_0\\approx 1.3247$ is the real root of $x^3=x+1$, and is the smallest Pisot-Vijayaraghavan number.\n\nIn [EJK90] Erdő, Joó, and Komornik prove that any Pisot-Vijayaraghavan number cannot have this property, and also prove that, for any $10\\]for all $m\\geq 1$, where $x_k^m$ is the set of those numbers which can be written as a finite sum $\\sum_{n\\geq 0}c_nq^n$ for some $c_n\\in \\{0,\\ldots,m\\}$ (so that the sequence in the question is $x_k^1$). Erdős, Joó, and Schnitzer [EJS96] improved this to show that, if $10.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#1096: [EJK90][GWNT91]" }, { "number": 1097, "url": "https://www.erdosproblems.com/1097", "status": "open", "prize": "no", "tags": [ "number theory", "additive combinatorics" ], "oeis": [], "formalized": "yes", "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$?In particular, are there always $O(n^{3/2})$ many such $d$?", "commentary": "A problem Erdős posed in the 1989 problem session of Great Western Number Theory.\n\nHe states that Erdős and Ruzsa gave an explicit construction which achieved $n^{1+c}$ for some $c>0$, and Erdős and Spencer gave a probabilistic proof which achieved $n^{3/2}$, and speculated this may be the best possible.\n\nIn the comment section, Chan has noticed that this problem is exactly equivalent to a sums-differences question of Bourgain [Bo99], introduced as an arithmetic path towards the Kakeya conjecture: find the smallest $c\\in [1,2]$ such that, for any finite sets of integers $A$ and $B$ and $G\\subseteq A\\times B$ we have\\[\\lvert A\\overset{G}{-}B\\rvert \\ll \\max(\\lvert A\\rvert,\\lvert B\\rvert, \\lvert A\\overset{G}{+}B\\rvert)^c\\](where, for example, $A\\overset{G}{+}B$ denotes the set of $a+b$ with $(a,b)\\in G$). \n\nThis is equivalent in the sense that the greatest exponent $c$ achievable for the main problem here is equal to the smallest constant achievable for the sums-differences question. The current best bounds known are thus\\[1.77898\\cdots \\leq c \\leq 11/6 \\approx 1.833.\\]The upper bound is due to Katz and Tao [KaTa99]. The lower bound is due to Lemm [Le15] (with a very small improvement found by AlphaEvolve [GGTW25]). \n\nThis resolves the second part of the question negatively, but the general question of the correct order of magnitude remains open.\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#1097: [GWNT89]" }, { "number": 1100, "url": "https://www.erdosproblems.com/1100", "status": "open", "prize": "no", "tags": [ "number theory", "divisors" ], "oeis": [ "A325864" ], "formalized": "no", "statement": "If $1=d_1<\\cdots0$ and sufficiently large $x$,\\[\\max_{n \\exp((\\log\\log x)^{2-\\epsilon}).\\]Erdős and Simonovits (see [Er81h]) proved\\[(2^{1/2}+o(1))^k < g(k) < (2-c)^k\\]for some constant $c>0$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#1100: [ErHa78][Er81h,p.173]" }, { "number": 1101, "url": "https://www.erdosproblems.com/1101", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "yes", "statement": "If $u=\\{u_10$, if $x$ is sufficiently large then\\[\\max_{a_k (1+o(1))t_x \\prod_{i}\\left(1-\\frac{1}{u_i}\\right)^{-1}.\\]The strong form of [208] is asking whether if $u_i=p_i^2$, the sequence of prime squares, is good.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 19 October 2025. View history", "references": "#1101: [Er81h,p.178]" }, { "number": 1103, "url": "https://www.erdosproblems.com/1103", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A392164" ], "formalized": "no", "statement": "Let $A$ be an infinite sequence of integers such that every $n\\in A+A$ is squarefree. How fast must $A$ grow?", "commentary": "Erdős notes there exists such a sequence which grows exponentially, but does not expect such a sequence of polynomial growth.\n\nIn [Er81h] he asked whether there is an infinite sequence of integers $A$ such that, for every $a\\in A$ and prime $p$, if\\[a\\equiv t\\pmod{p^2}\\]then $1\\leq t 0.24j^{4/3}$ for all $j$, and further that there exists such a sequence (furthermore with squarefree terms) such that\\[a_j < \\exp(5j/\\log j)\\]for all large $j$. A superior lower bound of $a_j \\gg j^{15/11-o(1)}$ had earlier been found by Konyagin [Ko04] when considering the finite case [1109].\n\nThey also obtain further results for the generalisation from squarefree to $k$-free integers, and also replacing $A+A$ with $A\\cup (A+A)\\cup(A+A+A)$. \n\nSee also [1109] for the finite analogue of this problem.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 December 2025. View history", "references": "#1103: [Er81h,p.180]" }, { "number": 1104, "url": "https://www.erdosproblems.com/1104", "status": "open", "prize": "no", "tags": [ "graph theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.", "commentary": "The best bounds available are\\[(1-o(1))(n/\\log n)^{1/2}\\leq f(n) \\leq (2+o(1))(n/\\log n)^{1/2}.\\]The upper bound is due to Davies and Illingworth [DaIl22], the lower bound follows from a construction of Hefty, Horn, King, and Pfender [HHKP25].\n\nOne can ask a similar question for the maximum possible chromatic number of a triangle-free graph on $m$ edges. Let this be $g(m)$. Davies and Illingworth [DaIl22] prove\\[g(m) \\leq (3^{5/3}+o(1))\\left(\\frac{m}{(\\log m)^2}\\right)^{1/3}.\\]Kim [Ki95] gave a construction which implies $g(m) \\gg (m/(\\log m)^2)^{1/3}$.\n\nThe function $f(n)$ is the inverse to the function $h_3(k)$ considered in [1013].\n\nA generalisation of $f(n)$ is considered in [920].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 21 January 2026. View history", "references": "#1104: [Er67c]" }, { "number": 1106, "url": "https://www.erdosproblems.com/1106", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A194259", "A194260" ], "formalized": "yes", "statement": "Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of\\[\\prod_{1\\leq k\\leq n}p(k).\\]Does $F(n)\\to \\infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?", "commentary": "Asked by Erdős at Oberwolfach in 1986. Schinzel noted in the Oberwolfach problem book that $F(n)\\to \\infty$ follows from the asymptotic formula for $p(n)$ and a result of Tijdeman [Ti73]. This is not obvious; details are given in a paper of Erdős and Ivić (see page 69 of [ErIv90]).\n\nSchinzel and Wirsing [ScWi87] have proved $F(n) \\gg \\log n$. \n\nOno [On00] has proved that every prime divides $p(n)$ for some $n\\geq 1$ (indeed this holds, for any fixed prime, for a positive density set of $n$).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 16 November 2025. View history", "references": "#1106: [Ob1]" }, { "number": 1107, "url": "https://www.erdosproblems.com/1107", "status": "open", "prize": "no", "tags": [ "number theory", "powerful" ], "oeis": [ "A056828", "A392342", "A392343" ], "formalized": "yes", "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?", "commentary": "Given in the 1986 Oberwolfach problem book as a problem of Erdős and Ivić.\n\nThis is true when $r=2$, as proved by Heath-Brown [He88] (see [941]). \n\nSee [940] for the problem of which integers are the sum of at most $r$ many $r$-powerful numbers.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 18 November 2025. View history", "references": "#1107: [Ob1]" }, { "number": 1108, "url": "https://www.erdosproblems.com/1108", "status": "open", "prize": "no", "tags": [ "number theory", "factorials" ], "oeis": [ "A051761", "A115645", "A025494" ], "formalized": "yes", "statement": "Let\\[A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}.\\]If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?", "commentary": "Asked by Erdős at Oberwolfach in 1988. It is open even whether there are infinitely many squares of the form $1+n!$ (see [398]).\n\nThis was motivated in part by a problem of Mahler which he discussed with Erdős a few days before his death in 1988: if $k\\geq 5$ and\\[A_k= \\left\\{ \\sum_{n\\in S}k^n : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}\\]then does $A_k$ contain only finitely many squares? Mahler showed that there are infinitely many squares in $A_k$ for $k\\leq 4$, and found only one square for $k\\geq 5$, namely\\[1+7+7^2+7^3=400.\\]Brindza and Erdős [BrEr91] proved that, for any $r$, if $n_1!+\\cdots+n_r!$ is powerful then $n_1\\ll_r 1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 04 November 2025. View history", "references": "#1108: [Ob1]" }, { "number": 1109, "url": "https://www.erdosproblems.com/1109", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [ "A392164", "A392165" ], "formalized": "no", "statement": "Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that every $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?", "commentary": "First studied by Erdős and Sárközy [ErSa87], who proved\\[\\log N \\ll f(N) \\ll N^{3/4}\\log N,\\]and guessed the lower bound is nearer the truth. Sárközy [Sa92c] extended this to consider the case of $A+B$ and also looking for sumsets which are $k$-power-free.\n\nGyarmati [Gy01] gave an alternative proof of $f(N)\\gg \\log N$, and also gave new bounds for the case of $A+B$. Konyagin [Ko04] improved this to\\[ \\log\\log N(\\log N)^2\\ll f(N) \\ll N^{11/15+o(1)}.\\]The infinite analogue of this problem is [1103]. (In particular upper bounds for this $f(N)$ directly imply lower bounds for the size of the $a_j$ considered there.)\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 03 December 2025. View history", "references": "#1109: [ErSa87]" }, { "number": 1110, "url": "https://www.erdosproblems.com/1110", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other. If $\\{p,q\\}\\neq \\{2,3\\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?", "commentary": "A problem of Erdős and Lewin [ErLe96], who proved that there are finitely many non-representable numbers if and only if $\\{p,q\\}=\\{2,3\\}$.\n\nIndeed, in [Er92b] Erdős wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. He wrote 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.' \n\nThis simple proof is as follows: one proves the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even: if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).\n\nYu and Chen [YuCh22] prove that the set of representable numbers has density zero whenever $q>3$ or $q=3$ and $p>6$ or $q=2$ and $p>10$. They also prove that there are infinitely many coprime non-representable numbers if $q>3$ or $q=3$ and $p\\neq 5$ or $q=2$ and $p\\not\\in \\{3,5,9\\}$.\n\nErdős and Lewin [ErLe96] also asked whether all large integers $n$ can be written as a sum of $2^k3^l$, none of which divide another, each of which is $>f(n)$ for some $f(n)\\to \\infty$. Let $f(n)$ be the fastest growing such $f(n)$. Yu and Chen [YuCh22] proved\\[\\frac{n}{(\\log n)^{\\log_23}}\\ll f(n) \\ll \\frac{n}{\\log n}.\\]Yang and Zhao [YaZh25] improved the lower bound to $f(n)\\gg n/\\log n$. van Doorn has observed in the comments that a result of Blecksmith, McCallum, and Selfridge [BMS98] already implies\\[f(n)\\sim \\frac{\\log 2\\log 3}{2}\\frac{n}{\\log n}.\\]The case of three powers is the subject of [123], and see also [845] for more on the case $\\{p,q\\}=\\{2,3\\}$. The problem [246] addresses the topic without the non-divisibility condition.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#1110: [ErLe96]" }, { "number": 1111, "url": "https://www.erdosproblems.com/1111", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.If $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)3$.\n\nNguyen, Scott, and Seymour [NSS24] prove that if $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)0$, and if $f(p)$ does not have very large values (in a certain technical sense).\n\nSee also [491].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#1122: [Er46]" }, { "number": 1131, "url": "https://www.erdosproblems.com/1131", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials" ], "oeis": [], "formalized": "no", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\neq k$.What is the minimal value of\\[I(x_1,\\ldots,x_n)=\\int_{-1}^1 \\sum_k \\lvert l_k(x)\\rvert^2\\mathrm{d}x?\\]In particular, is it true that\\[\\min I =2-(1+o(1))\\frac{1}{n}?\\]", "commentary": "Erdős first conjectured this minimum was achieved by taking the $x_i$ to be the roots of the integral of the Legendre polynomial, since Fejer [Fe32] had earlier shown these to be minimisers of\\[\\max_{x\\in [-1,1]}\\sum_k \\lvert l_k(x)\\rvert^2.\\]This was disproved by Szabados [Sz66] for every $n>3$. \n\nErdős, Szabados, Varma, and Vértesi [ESVV94] proved that\\[2-O\\left(\\frac{(\\log n)^2}{n}\\right)\\leq \\min I\\leq 2-\\frac{2}{2n-1}\\]where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1131: [Er61,p.67][ESVV94][Er95e][Va99,2.45]" }, { "number": 1132, "url": "https://www.erdosproblems.com/1132", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials" ], "oeis": [], "formalized": "no", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\neq k$.Let $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let\\[L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert,\\]where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.Must there exist $x\\in (-1,1)$ such that\\[L_n(x) >\\frac{2}{\\pi}\\log n-O(1)\\]for infinitely many $n$?Is it true that\\[\\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi}\\]for almost all $x\\in (-1,1)$?", "commentary": "A result of Bernstein [Be31] implies that the set of $x\\in(-1,1)$ for which\\[\\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi}\\]is everywhere dense.\n\nErdős [Er61c] proved that, for any fixed $x_1,\\ldots,x_n\\in [-1,1]$,\\[\\max_{x\\in [-1,1]}\\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert>\\frac{2}{\\pi}\\log n-O(1).\\]Tao [Ta26b] has proved that for any function $\\omega(n)$ which tends to infinity with $n$ there exists a dense set of $x\\in (-1,1)$ such that\\[L_n(x)\\geq \\frac{2}{\\pi}\\log n-\\omega(n)\\]for infinitely many $n$. Tao also notes that it is unclear in the question of Erdős whether the constant in the $O(1)$ term may depend on $x$.\n\nSee also [1129] for more on $L_n(x)$, and also [1153].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 01 April 2026. View history", "references": "#1132: [Er67,p.68][Va99,2.43]" }, { "number": 1133, "url": "https://www.erdosproblems.com/1133", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials" ], "oeis": [], "formalized": "no", "statement": "Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.For any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many $1\\leq i\\leq n$, then\\[\\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C.\\]", "commentary": "Erdős proved that, for any $C>0$, there exists $\\epsilon>0$ such that if $n$ is sufficiently large and $m=\\lfloor (1+\\epsilon)n\\rfloor$ then for any $x_1,\\ldots,x_m\\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\\lvert P(x_i)\\rvert\\leq 1$ for $1\\leq i\\leq m$ and\\[\\max_{x\\in [-1,1]}\\lvert P(x)\\rvert>C.\\]The conjectured statement would also imply this, but Erdős in [Er67] says he could not even prove it for $m=n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 31 December 2025. View history", "references": "#1133: [Er67,p.72]" }, { "number": 1135, "url": "https://www.erdosproblems.com/1135", "status": "open", "prize": "$500", "tags": [ "number theory" ], "oeis": [ "A006370", "A008908" ], "formalized": "yes", "statement": "Define $f:\\mathbb{N}\\to \\mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\\frac{3n+1}{2}$ if $n$ is odd.Given any integer $m\\geq 1$ does there exist $k\\geq 1$ such that $f^{(k)}(m)=1$?", "commentary": "The infamous Collatz conjecture. For a detailed discussion of the history and theory surrounding this problem we refer to the overview by Lagarias [La10].\n\nThis is not a problem due to Erdős; it was first devised by Collatz before 1952. Erdős referred to this problem on several occasions as 'hopeless'. As Lagarias [La16] notes, the closest Erdős ever came to working on problems of this nature is the theorem described in the remarks to [1134]. \n\nIt is often claimed that Erdős offered \\$500 for a solution to this problem; this claim originated in a survey article by Lagarias [La85].\n\nLagarias reported, in personal communication, that this came from a conversation he had with Erdős and Graham around 1983, in which Graham asked Erdős to make an estimate of what value Erdős would put the problem on his prize scale, to which Erdős replied \\$500. Therefore, strictly speaking, Erdős never offered \\$500 specifically as a prize, but we include this prize value here for comparing those problems which Erdős rated as 'prize problems'.\n\nThis is Problem E16 in Guy's collection [Gu04], in which Guy quotes Erdős as saying \"Mathematics may not be ready for such problems\".\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 12 January 2026. View history", "references": "#1135: [La85][Er97e,p.537][La16]" }, { "number": 1137, "url": "https://www.erdosproblems.com/1137", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [ "A083550", "A005250" ], "formalized": "yes", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\\[\\frac{\\max_{n1$. If $d=\\max_{p_n105$) such that $n-2^k$ is prime for all $1<2^k0$.\n\nThis is discussed in problem A19 of Guy's collection [Gu04]. There is also further discussion on the Prime Puzzles website.\n\nErdős made the stronger conjecture (see [236]) that that number of $1<2^k2$.", "commentary": "In [Va99] it is reported that Erdős and Selfridge found 'the exact bound' when $2<\\alpha<3$, and that 'if $\\alpha>3$ then very little is known'. No reference is given, and I cannot find a relevant paper of Erdős and Selfridge.\n\nSee also [970].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1143: [Va99,1.8]" }, { "number": 1144, "url": "https://www.erdosproblems.com/1144", "status": "open", "prize": "no", "tags": [ "number theory", "probability" ], "oeis": [], "formalized": "no", "statement": "Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random. Is it true that\\[\\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N}}=\\infty\\]with probability $1$?", "commentary": "This model of a random multiplicative function is sometimes called a Rademacher function, although this is sometimes reserved for a merely multiplicative function (which is $0$ on non-squarefree integers). See [520] for the partial sums of this alternative model.\n\nIt should also be compared to another popular model of random completely multiplicative functions, Steinhaus functions, which have $f(p)$ uniformly distributed over the unit circle.\n\nAtherfold [At25] has proved that, almost surely,\\[\\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log N)^{1+o(1)}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 January 2026. View history", "references": "#1144: [Va99,1.11]" }, { "number": 1145, "url": "https://www.erdosproblems.com/1145", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "additive basis" ], "oeis": [], "formalized": "yes", "statement": "Let $A=\\{1\\leq a_1d_s(B)$ for every $B\\subset \\mathbb{N}$ with $00$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\\pm 1$,\\[\\max_{\\lvert z\\rvert=1}\\lvert P(z)\\rvert > (1+c)\\sqrt{n}?\\]", "commentary": "In other words, does there exist an 'ultraflat' polynomial with coefficients $\\pm 1$. The answer is yes if the coefficients can take any values on the unit circle (see [230]).\n\nThe lower bound\\[\\max_{\\lvert z\\rvert=1}\\lvert P(z)\\rvert\\geq \\sqrt{n}\\]is trivial from Parseval's theorem.\n\nA weaker 'flatness' question is the subject of [228].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1150: [Ha74,4.31][Va99,2.36]" }, { "number": 1151, "url": "https://www.erdosproblems.com/1151", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials" ], "oeis": [], "formalized": "no", "statement": "Given $a_1,\\ldots,a_n\\in [-1,1]$ let\\[\\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i)\\ell_i(x)\\]be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\\leq i\\leq n$ (that is, the Lagrange interpolation polynomial).Let $a_i$ be the set of Chebyshev nodes. Prove that, for any closed $A\\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\\mathcal{L}^nf(x)$.", "commentary": "This is as the problem is given in [Va99], but I am unclear exactly what is intended here - is this meant for fixed, arbitrary, $x\\in [-1,1]$?\n\nErdős [Er41] proved that, if $x=\\cos(\\pi p/q)$ for some odd integers $p,q\\geq 1$, then there is a continuous function $f$ such that\\[\\lim_{n\\to \\infty}\\mathcal{L}^nf(x)=\\infty,\\]where the limit is taken over the sequence of Chebyshev nodes as $n\\to \\infty$. In [Er43] he claims (without proof) that for any closed set $A$ there exists a continuous $f$ such that the limit points of $\\mathcal{L}^nf(x)$ is $A$ (for specific $x$ of this shape).\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1151: [Va99,2.41]" }, { "number": 1152, "url": "https://www.erdosproblems.com/1152", "status": "open", "prize": "no", "tags": [ "analysis", "polynomials" ], "oeis": [], "formalized": "no", "statement": "For $n\\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\\ldots,x_{nn}\\in [-1,1]$. Let $\\epsilon=\\epsilon(n)\\to 0$. Does there always exist a continuous function $f:[-1,1]\\to \\mathbb{R}$ such that if $p_n$ is a sequence of polynomials, with degrees $\\deg p_n<(1+\\epsilon(n))n$, such that $p_n(x_{kn})=f(x_{kn})$ for all $1\\leq k\\leq n$, then $p_n(x)\\not\\to f(x)$ for almost all $x\\in [-1,1]$?", "commentary": "Erdős, Kroó, and Szabados [EKS89] proved that, if $\\epsilon>0$ is fixed and does not $\\to 0$, then there exist sequences $x_{ij}$ such that, for any continuous function $f$, there exists a sequence of polynomials $p_n$, with degrees $\\deg p_n<(1+\\epsilon)n$, such that $p_n(x_{kn})=f(x_{kn})$ for all $1\\leq k\\leq n$, and $p_n(x)\\to f(x)$ uniformly for all $x\\in [-1,1]$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1152: [Va99,2.42]" }, { "number": 1154, "url": "https://www.erdosproblems.com/1154", "status": "not disprovable", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Does there exist, for every $\\alpha \\in [0,1]$, a ring or field in $\\mathbb{R}$ with Hausdorff dimension $\\alpha$?", "commentary": "Erdős and Volkmann [ErVo66] proved that, for any $\\alpha\\in[0,1]$, there exists a group of real numbers of Hausdorff dimension $\\alpha$.\n\nFalconer [Fa84] proved that any subring with Hausdorff dimension $\\alpha \\in(1/2,1)$ cannot be a Borel or Suslin set. Edgar and Miller [EdMi01] proved that any real closed analytic subfield of $\\mathbb{R}$ has Hausdorff dimension either $0$ or $1$. Later the same authors [EdMi03] proved that any subring of $\\mathbb{R}$ which is Borel or analytic either has Hausdorff dimension $0$ or is equal to $\\mathbb{R}$.\n\nMauldin [Ma16b] proved that subfields of $\\mathbb{R}$ exist with Hausdorff dimension any $\\alpha \\in [0,1]$ assuming the continuum hypothesis.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 25 January 2026. View history", "references": "#1154: [Er79h,p.119][Va99,2.48]" }, { "number": 1155, "url": "https://www.erdosproblems.com/1155", "status": "open", "prize": "no", "tags": [ "graph theory" ], "oeis": [], "formalized": "no", "statement": "Construct a random graph on $n$ vertices in the following way: begin with the complete graph $K_n$. At each stage, choose uniformly a random triangle in the graph and delete all the edges of this triangle. Repeat until the graph is triangle-free.Describe the typical parameters and structure of such a graph. In particular, if $f(n)$ is the number of edges remaining, then is it true that\\[\\mathbb{E}f(n)\\asymp n^{3/2}\\]and that $f(n) \\ll n^{3/2}$ almost surely?", "commentary": "A problem of Bollobás and Erdős, described in [Va99] as 'motivated by the task of generating a random triangle-free graph'. In [Bo98] it says they asked this at the 'Quo Vadis, Graph Theory?' conference in Fairbanks, Alaska, in 1990, 'while admiring the playful bears'.\n\nGrable [Gr97] proved that, for every $\\epsilon>0$,\\[\\mathbb{P}(f(n)>n^{7/4+\\epsilon})\\to 0.\\]Bohman, Frieze, and Lubetzky [BFL15] proved that $f(n)=n^{3/2+o(1)}$ almost surely - in other words, for every $\\epsilon>0$,\\[\\mathbb{P}(n^{3/2-\\epsilon}n^c$. This was improved to any $c<1/2$ by Heckel and Riordan [HeRi23].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 27 January 2026. View history", "references": "#1156: [AlSp92][Va99,3.6]" }, { "number": 1157, "url": "https://www.erdosproblems.com/1157", "status": "open", "prize": "no", "tags": [ "hypergraphs", "turan number" ], "oeis": [], "formalized": "no", "statement": "Let $t,k,r\\geq 2$. Let $\\mathcal{F}$ be the family of all $r$-uniform hypergraphs with $k$ vertices and $s$ edges. Determine\\[\\mathrm{ex}_r(n,\\mathcal{F}).\\]", "commentary": "This is a very difficult and general question, and many partial results are known. The paper of Brown, Erdős, and Sós [BES73] contains many of them, in particular the general lower bound, for all $k>r$ and $s>1$,\\[\\mathrm{ex}_r(n,\\mathcal{F})\\gg_{k,s} n^{\\frac{rs-k}{s-1}}.\\]A general conjecture of Brown, Erdős, and Sós is that, for all $r>t\\geq 2$ and $s\\geq 3$,\\[\\mathrm{ex}_t(n,\\mathcal{F})=o(n^t)\\]whenever $k\\geq (r-t)s+t+1$. The case $t=2$ is the subject of [1178].\n\nThe case $s=r=3$ and $k=6$ is the subject of [716]. The case of $r=3$ and $k=s+2$ is the subject of [1076].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 January 2026. View history", "references": "#1157: [BES73][Va99,3.64]" }, { "number": 1158, "url": "https://www.erdosproblems.com/1158", "status": "open", "prize": "no", "tags": [ "hypergraphs", "turan number" ], "oeis": [], "formalized": "no", "statement": "Let $K_{t}(r)$ be the complete $t$-partite $t$-uniform hypergraph with $r$ vertices in each class. Is it true that\\[\\mathrm{ex}_t(n,K_t(r)) \\geq n^{t-r^{1-t}-o(1)}\\]for all $t,r$?", "commentary": "Erdős [Er64f] proved that\\[n^{t-O(r^{1-t})}\\leq \\mathrm{ex}_t(n,K_t(r)) \\ll n^{t-r^{1-t}}.\\]This is only known when $t=2$ and $2\\leq r\\leq 3$. The case $t=2$ is [714].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1158: [Va99,3.65]" }, { "number": 1159, "url": "https://www.erdosproblems.com/1159", "status": "open", "prize": "no", "tags": [ "combinatorics" ], "oeis": [], "formalized": "no", "statement": "Determine whether there exists a constant $C>1$ such that the following holds.Let $P$ be a finite projective plane. Must there exist a set of points $S$ such that $1\\leq \\lvert S\\cap \\ell\\rvert \\leq C$ for all lines $\\ell$?", "commentary": "A set which meets all lines at once is called a blocking set. In [Er81] Erdős asks the stronger question of whether this is true for all pairwise balanced block designs.\n\nErdős, Silverman, and Stein [ESS83] proved this is true with $\\lvert S\\cap\\ell \\rvert\\ll \\log n$ for all lines $\\ell$ (where $n$ is the order of the projective plane).\n\nSee also [664] for a stronger question. This problem is mentioned after Problem 68 on Green's open problems list.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 10 April 2026. View history", "references": "#1159: [Va99,4.70]" }, { "number": 1160, "url": "https://www.erdosproblems.com/1160", "status": "open", "prize": "no", "tags": [ "group theory" ], "oeis": [ "A000001" ], "formalized": "no", "statement": "Let $g(n)$ denote the number of groups of order $n$. If $n\\leq 2^m$ then $g(n)\\leq g(2^m)$.", "commentary": "This is listed as an open problem (Question 22.16) in [BNV07], which reports it as a 'quite natural conjecture, whose origin we have been unable to trace satisfactorily. We have heard it attributed at various times to various people, such as Paul Erdős and Graham Higman.'\n\nQuestion 22.18 of [BNV07] suggests the even stronger conjecture\\[\\sum_{n<2^m}g(n) \\leq g(2^m)\\]for all sufficiently large $m$ (perhaps even as soon as $m\\geq 7$).\n\nPantelidakis [Pa03] proved that the original conjecture is true if $n$ is odd and $m\\geq 3619$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 January 2026. View history", "references": "#1160: [Va99,5.71]" }, { "number": 1162, "url": "https://www.erdosproblems.com/1162", "status": "open", "prize": "no", "tags": [ "group theory" ], "oeis": [], "formalized": "no", "statement": "Give an asymptotic formula for the number of subgroups of $S_n$. Is there a statistical theorem on their order?", "commentary": "A problem of Erdős and Turán.\n\nLet $f(n)$ count the number of subgroups of $S_n$. Pyber [Py93] proved that\\[\\log f(n) \\asymp n^2.\\]Roney-Dougal and Tracey [RoTr25] have proved that\\[\\log f(n)=\\left(\\frac{1}{16}+o(1)\\right)n^2.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1162: [Va99,5.73]" }, { "number": 1163, "url": "https://www.erdosproblems.com/1163", "status": "open", "prize": "no", "tags": [ "group theory" ], "oeis": [], "formalized": "no", "statement": "Describe (by statistical means) the arithmetic structure of the orders of subgroups of $S_n$.", "commentary": "This is given in [Va99] as a problem of Erdős and Turán. I have reproduced the problem verbatim; I am not entirely sure what it is asking for.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1163: [Va99,5.74]" }, { "number": 1167, "url": "https://www.erdosproblems.com/1167", "status": "open", "prize": "no", "tags": [ "set theory", "probability" ], "oeis": [], "formalized": "no", "statement": "Let $r\\geq 2$ be finite and $\\lambda$ be an infinite cardinal. Let $\\kappa_\\alpha$ be cardinals for all $\\alpha<\\gamma$. Is it true that\\[2^\\lambda \\to (\\kappa_\\alpha+1)_{\\alpha<\\gamma}^{r+1}\\]implies\\[\\lambda \\to (\\kappa_\\alpha)_{\\alpha<\\gamma}^{r}?\\]Here $+$ means cardinal addition, so that $\\kappa_\\alpha+1=\\kappa_\\alpha$ if $\\kappa_\\alpha$ is infinite.", "commentary": "A problem of Erdős, Hajnal, and Rado.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1167: [Va99,7.79]" }, { "number": 1168, "url": "https://www.erdosproblems.com/1168", "status": "open", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Prove that\\[\\aleph_{\\omega+1}\\not\\to (\\aleph_{\\omega+1}, 3,\\ldots,3)_{\\aleph_0}^2\\]without assuming the generalised continuum hypothesis.", "commentary": "A problem of Erdős, Hajnal, and Rado.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1168: [Va99,7.80]" }, { "number": 1169, "url": "https://www.erdosproblems.com/1169", "status": "not disprovable", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Is it true that, for all finite $k<\\omega$,\\[\\omega_1^2 \\not\\to (\\omega_1^2, 3)^2?\\]", "commentary": "A problem of Erdős and Hajnal. Hajnal [Ha71] proved this is true assuming the continuum hypothesis.\n\nSee also [592] for a similar problem concerning countable ordinals.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 25 January 2026. View history", "references": "#1169: [Va99,7.85]" }, { "number": 1170, "url": "https://www.erdosproblems.com/1170", "status": "open", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Is it consistent that\\[\\omega_2\\to (\\alpha)_2^2\\]for every $\\alpha <\\omega_2$?", "commentary": "Laver [La82] proved the consistency of $\\omega_2\\to (\\omega_12+1,\\alpha)^2$ for all $\\alpha<\\omega_2$. Foreman and Hajnal [FoHa03] proved the consistency of $\\omega_2\\to (\\omega_1^2+1,\\alpha)^2$ for all $\\alpha<\\omega_2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1170: [Va99,7.86]" }, { "number": 1171, "url": "https://www.erdosproblems.com/1171", "status": "open", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Is it true that, for all finite $k<\\omega$,\\[\\omega_1^2\\to (\\omega_1\\omega, 3,\\ldots,3)_{k+1}^2?\\]", "commentary": "Baumgartner [Ba89b] proved that, assuming a form of Martin's axiom, that $\\omega_1\\omega\\to (\\omega_1\\omega, 3)^2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 January 2026. View history", "references": "#1171: [Va99,7.84]" }, { "number": 1172, "url": "https://www.erdosproblems.com/1172", "status": "open", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is consistent with the generalised continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2,\\]or even $\\omega_2 \\to (\\xi)_2^2$ for all $\\xi<\\omega_2$.", "commentary": "A problem of Erdős and Hajnal. The Erdős-Rado partition theorem [ErRa56] states that\\[(2^{\\kappa})^+ \\to (\\kappa^++1)_\\kappa^2\\]for every infinite cardinal $\\kappa$.\n \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 11 April 2026. View history", "references": "#1172: [ErHa74,p.272][Va99,7.87]" }, { "number": 1173, "url": "https://www.erdosproblems.com/1173", "status": "open", "prize": "no", "tags": [ "set theory", "combinatorics" ], "oeis": [], "formalized": "no", "statement": "Assume the generalised continuum hypothesis. Let\\[f: \\omega_{\\omega+1}\\to [\\omega_{\\omega+1}]^{\\leq \\aleph_\\omega}\\]be a set mapping such that\\[\\lvert f(\\alpha)\\cap f(\\beta)\\rvert <\\aleph_\\omega\\]for all $\\alpha\\neq \\beta$. Does there exist a free set of cardinality $\\aleph_{\\omega+1}$?", "commentary": "A problem of Erdős and Hajnal.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 25 January 2026. View history", "references": "#1173: [Ko25b,Problem 35][Va99,7.88]" }, { "number": 1174, "url": "https://www.erdosproblems.com/1174", "status": "not disprovable", "prize": "no", "tags": [ "set theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_3$?Does there exist a graph $G$ with no $K_{\\aleph_1}$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_{\\aleph_0}$?", "commentary": "A problem of Erdős and Hajnal. Shelah proved that a graph with either property can consistently exist.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1174: [Va99,7.91]" }, { "number": 1175, "url": "https://www.erdosproblems.com/1175", "status": "open", "prize": "no", "tags": [ "set theory", "chromatic number" ], "oeis": [], "formalized": "no", "statement": "Let $\\kappa$ be an uncountable cardinal. Must there exist a cardinal $\\lambda$ such that every graph with chromatic number $\\lambda$ contains a triangle-free subgraph with chromatic number $\\kappa$?", "commentary": "Shelah proved that a negative answer is consistent if $\\kappa=\\lambda=\\aleph_1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1175: [Va99,7.92]" }, { "number": 1176, "url": "https://www.erdosproblems.com/1176", "status": "not disprovable", "prize": "no", "tags": [ "set theory", "chromatic number" ], "oeis": [], "formalized": "yes", "statement": "Let $G$ be a graph with chromatic number $\\aleph_1$. Is it true that there is a colouring of the edges with $\\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?", "commentary": "A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 24 January 2026. View history", "references": "#1176: [Va99,7.93]" }, { "number": 1177, "url": "https://www.erdosproblems.com/1177", "status": "open", "prize": "no", "tags": [ "set theory", "chromatic number", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\\kappa$ not containing $G$.If $F_G(\\aleph_1)$ is not empty then there exists $X\\in F_G(\\aleph_1)$ of cardinality at most $2^{2^{\\aleph_0}}$.If both $F_G(\\aleph_1)$ and $F_H(\\aleph_1)$ are non-empty then $F_G(\\aleph_1)\\cap F_H(\\aleph_1)$ is non-empty.If $\\kappa,\\lambda$ are uncountable cardinals and $F_G(\\kappa)$ is non-empty then $F_G(\\lambda)$ is non-empty.", "commentary": "A problem of Erdős, Galvin, and Hajnal.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 23 January 2026. View history", "references": "#1177: [Va99,7.94]" }, { "number": 1178, "url": "https://www.erdosproblems.com/1178", "status": "open", "prize": "no", "tags": [ "graph theory", "hypergraphs" ], "oeis": [], "formalized": "no", "statement": "For $r\\geq 3$ let $d_r(e)$ be the minimal $d$ such that\\[\\mathrm{ex}_r(n,\\mathcal{F})=o(n^2),\\]where $\\mathcal{F}$ is the family of $r$-uniform hypergraphs on $d$ vertices with $e$ edges.Prove that\\[d_r(e)=(r-2)e+3\\]for all $r,e\\geq 3$.", "commentary": "A conjecture of Brown, Erdős, and Sós [BES73], who proved that $d_r(e)\\geq (r-2)e+3$ (see also [1076]).\n\nRuzsa and Szemerédi [RuSz78] proved $d_3(3)=6$ (see [716]).\n\nErdős, Frankl, and Rödl [EFR86] proved\\[d_r(3)= (r-2)3+3\\]for all $r\\geq 3$. Sárközy and Selkow [SaSe05] proved\\[d_r(e) \\leq (r-2)e+2+\\lfloor \\log_2 e\\rfloor\\]for all $r,e\\geq 3$. Solymosi and Solymosi [SoSo17] proved that $d_3(10)\\leq 14$. Conlon, Gishboliner, Levanzov, and Shapira [CGLS23] proved\\[d_3(e)\\leq e+O\\left(\\frac{\\log e}{\\log\\log e}\\right)\\]for all $e\\geq 3$.\n\nIn [Er75b] Erdős even asks whether, if $\\mathcal{F}$ is the family of $3$-uniform hypergraphs on $k$ vertices with $k-3$ edges,\\[\\mathrm{ex}_3(n,\\mathcal{F})\\asymp n r_{k-3}(n),\\]where $r_{k-3}(n)$ is the maximal size of a subset of $\\{1,\\ldots,n\\}$ that does not contain a non-trivial arithmetic progression of length $k-3$. He states that Ruzsa has proved the lower bound for $k=6,7,8$.\n\nSee [1157] for the general case.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 26 January 2026. View history", "references": "#1178: [BES73][Er75b][Er81]" }, { "number": 1181, "url": "https://www.erdosproblems.com/1181", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\\[q(n,\\log n)<(1-c)(\\log n)^2?\\]", "commentary": "The upper bound $q(n,\\log n)\\leq (1+o(1))(\\log n)^2$ follows from observing that the primorial of $q(n,k)$ is at most $\\prod_{1\\leq i\\leq k}(n+i)$.\n\nProbabilistic heuristics described by Tao in the comments of [457] suggest that\\[q(n,\\log n)\\ll \\frac{\\log\\log n}{\\log\\log\\log n}{\\log n}\\]for all $n$. \n\n[457] concerns lower bounds for $q(n,\\log n)$ (and in particular the constructions described there proved that the above upper bound would be best possible).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 March 2026. View history", "references": "#1181: [Er79d,p.78]" }, { "number": 1182, "url": "https://www.erdosproblems.com/1182", "status": "open", "prize": "no", "tags": [ "graph theory", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\\[R(K_3,G)= 2n-1.\\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\\leq F(n)$ edges has\\[R(K_3,G)= 2n-1.\\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\\to \\infty$?", "commentary": "A problem of Burr, Erdős, Faudree, Rousseau, and Schelp. \n\nIt is trivial that $f(n)\\geq F(n)$, and that $R(K_3,G)\\geq 2n-1$ for any connected graph $G$ on $n$ vertices. By a result of Chavátal $R(K_3,T)=2n-1$ whenever $T$ is a tree on $n$ vertices, whence $F(n)\\geq n-1$.\n\nBurr, Erdős, Faudree, Roussea, and Schelp [BEFRS80] proved\\[\\frac{17n+1}{15}\\leq F(n)\\leq \\left(\\frac{27}{4}+o(1)\\right)n(\\log n)^2,\\]the lower bound for all $n\\geq 4$. The upper bound was improved to\\[F(n)\\leq 84n\\]by Brandt [Br96], who expects that for all large $n$ we have $2nn^{\\omega(n)}$ for some $\\omega(n)\\to \\infty$, but gave no reference.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1183: [Er78,p.39]" }, { "number": 1184, "url": "https://www.erdosproblems.com/1184", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such that $n=k^{\\alpha+o(1)}$, then\\[f(n,k)=(1-\\rho(\\alpha)+o(1))k,\\]where $\\rho$ is the Dickman function?", "commentary": "Erdős [Er76e] proved that, for every $\\alpha>1$, if $k$ is sufficiently large and $n>k^\\alpha-k$ then\\[f(n,k) > \\left(1-\\frac{1}{\\alpha}+c_\\alpha\\right)k\\]for some constant $c_\\alpha>0$, and if $1<\\alpha<2$ and $n\\leq k^\\alpha-k$ then\\[f(n,k) < (\\alpha-1+o(1))k.\\]He knew of no non-trivial bounds when $\\alpha \\geq 2$.\n\nRamachandra, Shorey, and Tijdeman [RST75b] proved that if $n>\\exp(c(\\log k)^2)$ for a constant $c>0$ then $f(n,k)\\geq k-\\pi(k)$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1184: [Er76e,p.272]" }, { "number": 1186, "url": "https://www.erdosproblems.com/1186", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [], "formalized": "no", "statement": "Let $\\delta_k$ be such that in any $2$-colouring of $\\{1,\\ldots,n\\}$ there exist at least $(\\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\\delta_k$.", "commentary": "In [Er80] Erdős said 'perhaps one can get an asymptotic formula' for $\\delta_k$, but this seems unlikely. Van der Waerden's theorem implies $\\delta_k\\gg_k 1$. The probabilistic method implies\\[\\delta_k \\leq \\frac{1}{(k-1)2^{k}}.\\]Parrilo, Robertson, and Saracino [PRS08] have proved\\[0.0511\\approx \\frac{1675}{32768}\\leq \\delta_3 \\leq \\frac{117}{2192}\\approx 0.0533,\\]and conjectured the upper bound is correct. They also report that Graham offered \\$100 for determining the value of $\\delta_3$ at a conference in 1999.\n\nIt is easier to study this quantity if we replace $\\{1,\\ldots,n\\}$ with a finite field $\\mathbb{F}_p$; let this analogue be denoted by $\\tilde{\\delta}_k$. A random colouring implies $\\tilde{\\delta}_k\\leq 1/2^{k}$.\n\nIt is straightforward to prove that $\\tilde{\\delta}_3=1/8$ (see, for example, [Wo10]). Improving on work of Cameron, Cilleruelo, and Serra [CCS07], Wolf [Wo10] proved\\[\\frac{1}{32}\\leq \\tilde{\\delta}_4\\leq \\left(1-\\frac{1}{259200}\\right)\\frac{1}{16}.\\]These bounds were improved by Lu and Peng [LuPe12] to\\[\\frac{7}{192}\\leq \\tilde{\\delta}_4\\leq \\frac{17}{300}.\\]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1186: [Er80,p.93]" }, { "number": 1188, "url": "https://www.erdosproblems.com/1188", "status": "open", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "no", "statement": "Call a set of distinct integers $11$) form an irreducible covering set?", "commentary": "This is subtly different from the (more usually studied) notion of minimal covering system: an irreducible covering set corresponds to the set of moduli of a minimal covering system, but not necessarily vice versa.\n\nThe divisors of $12$ (which are $>1$) are an example of an irreducible covering set. Sun [Su07] proved more generally that, for any odd prime $p$, the divisors (which are $>1$) of $2^{p-1}p$ form an irreducible covering set, settling the final question.\n\nSimpson [Si85] proved $n_k\\leq 2^{k-1}$.\n\nLet $I(k)$ count the number of irreducible covering sets of size $k$. Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST24] proved that\\[I(k) \\leq \\exp\\left((c+o(1))\\frac{k^{3/2}}{(\\log k)^{1/2}}\\right)\\]by proving that the right-hand side is an asymptotic for the number of minimal covering systems on $k$ moduli.\n\nIt is trivial that $\\sum\\frac{1}{n_i}>1$ for any irreducible covering set.\n\nSee also [1188].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1189: [Er80,p.95]" }, { "number": 1190, "url": "https://www.erdosproblems.com/1190", "status": "open", "prize": "no", "tags": [ "number theory", "covering systems" ], "oeis": [], "formalized": "no", "statement": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m0\\]for some $c>0$?", "commentary": "Erdős proved (see for example [HaRo66]) that if $A$ is an infinite Sidon set then\\[\\liminf_{x\\to \\infty} \\frac{\\lvert A\\cap [1,x]\\rvert}{x^{1/2}}(\\log x)^{1/2}\\leq c\\]for some constant $c>0$. In [Er80] he offered \\$1000 'for clearing up the problems' raised by this; he may have meant for finding the optimal function $f$ such that\\[\\liminf_{x\\to \\infty} \\frac{\\lvert A\\cap [1,x]\\rvert}{x^{1/2}}f(x)=0\\]for all infinite Sidon sets $A$.\n\nThe second question is a stronger form of [39]. See also [729] for the behaviour of the $\\limsup$. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1191: [Er80,p.98]" }, { "number": 1192, "url": "https://www.erdosproblems.com/1192", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "additive basis" ], "oeis": [], "formalized": "no", "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$.Does there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]for all $x$?", "commentary": "Erdős and Rényi proved by the probabilistic method that there exists a set $A$ such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]and\\[\\lvert A\\cap [1,x]\\rvert\\gg x^{1/r}\\]for all $x$. \n\nRuzsa [Ru90] proved that the answer is yes for $r=2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1192: [Er80,p.99]" }, { "number": 1194, "url": "https://www.erdosproblems.com/1194", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "additive basis", "sidon sets" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subset\\mathbb{N}$ be such that every integer $n\\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\\in A$. How fast must $a_n/n$ increase?", "commentary": "Such a set is called a perfect difference set; they can be constructed, for example, by a greedy approach (as detailed by Lev [Le04]). The greedy construction achieves $a_n\\ll n^3$.\n\nErdős [Er80] wrote it is 'easy to see' that\\[\\limsup_{n\\to \\infty}\\frac{a_n}{n}=\\infty.\\]Indeed, since $A$ must be a Sidon set, if $a_n \\leq f(n)n$ for all $n$, then all differences in $[1,x/f(x)]$ can be realised by differences from $A\\cap [1,x]$, and hence $\\lvert A\\cap [1,x]\\rvert^2 \\gg x/f(x)$. Erdős (see for example [HaRo66]) proved that for infinitely many $x$ $\\lvert A\\cap [1,x]\\rvert \\ll (x/\\log x)^{1/2}$, which is a contradiction for a suitable choice of $f$.\n\nThis argument proves that $a_n\\gg n\\log n$ for infinitely many $n$.\n\nCilleruelo and Nathanson [CiNa08] describe a method for construction dense perfect difference sets from dense Sidon sets, allowing many of the results for the latter to transfer to the former.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1194: [Er80,p.100]" }, { "number": 1195, "url": "https://www.erdosproblems.com/1195", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $S\\subset \\mathbb{R}$ be a set of infinite measure such that $x/y$ is never an integer for all distinct $x,y\\in S$.How fast can $\\lvert S\\cap (0,x)\\rvert$ tend to infinity?", "commentary": "Schmidt [Sc69] asked whether such a set exists. Haight [Ha70] and Szemerédi [Sz71] have independently constructed such a set.\n\nErdős [Er80] writes it is 'easy to see' that $\\lvert S\\cap (0,x)\\rvert=o(x)$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1195: [Er80,p.101]" }, { "number": 1196, "url": "https://www.erdosproblems.com/1196", "status": "open", "prize": "no", "tags": [ "number theory", "primitive sets" ], "oeis": [], "formalized": "no", "statement": "Is it true that, for any $x$, if $A\\subset [x,\\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then\\[\\sum_{a\\in A}\\frac{1}{a\\log a}< 1+o(1),\\]where the $o(1)$ term $\\to 0$ as $x\\to \\infty$?", "commentary": "A conjecture of Erdős, Sárközy, and Szemerédi. Lichtman [Li23] has proved that\\[\\sum_{a\\in A}\\frac{1}{a\\log a}< e^{\\gamma}\\frac{\\pi}{4}+o(1)\\approx 1.399+o(1).\\]Lichtman [Li20] proved that if $A$ is the set of all integers with exactly $k$ prime factors (so that $A\\subset [2^k,\\infty)$ and $A$ is a primitive set) then\\[\\sum_{a\\in A}\\frac{1}{a\\log a}\\geq 1+O(k^{-1/2+o(1)}),\\]and suggested that the true rate of decay may be $O(2^{-k})$. Gorodetsky, Lichtman, and Wong [GLW24] have proved that\\[\\sum_{a\\in A}\\frac{1}{a\\log a}= 1-(c+o(1))k^22^{-k}\\]where $c\\approx 0.0656$ is an explicit constant.\n\nSee also [164] for the case $x=1$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1196: [ESS68b][Er80,p.101]" }, { "number": 1197, "url": "https://www.erdosproblems.com/1197", "status": "open", "prize": "no", "tags": [ "analysis" ], "oeis": [], "formalized": "no", "statement": "Let $E\\subset (0,\\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\geq 1$ such that $x\\in \\frac{r}{n}\\cdot E$?", "commentary": "A problem of Haight. \n\nThis is trivially true if $E$ contains an interval $(a,b)$ with $a0$, for all large $n$, the interval $(\\frac{nx}{b},\\frac{nx}{a})$ has length $>1$ so contains at least one integer $r\\geq 1$.\n\nBuczolich and Mauldin [BuMa99] proved that there exists an open set $E\\subset (0,\\infty)$ and two intervals $I,J\\subset [1/2,1)$ such that, for all $x\\in I$, $x\\in \\frac{1}{n}\\cdot E$ for infinitely many $n\\geq 1$, and for almost all $x\\in J$, $x\\not\\in \\frac{1}{n}\\cdot E$ for all sufficiently large (depending on $x$) $n$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 06 April 2026. View history", "references": "#1197: [Er80,p.103]" }, { "number": 1198, "url": "https://www.erdosproblems.com/1198", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "If $\\mathbb{N}$ is $2$-coloured then must there exist an infinite set $A=\\{a_1<\\cdots\\}$ such that all expressions of the shape\\[\\prod_{i\\in S_1}a_i+\\cdots+\\prod_{i\\in S_k}a_i,\\]for disjoint $S_1,\\ldots,S_k$, are the same colour?", "commentary": "The case when all $S_i$ are singletons was a conjecture of Graham and Rothschild proved by Hindman [Hi74] (see [532]). Hindman [Hi80] proved the answer is no if we $7$-colour the integers.\n\nErdős wrote 'one would perhaps guess that the answer must be \"no\" but no counterexample is in sight'.\n\nSee also [172].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1198: [Er80,p.104]" }, { "number": 1199, "url": "https://www.erdosproblems.com/1199", "status": "open", "prize": "no", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [], "formalized": "no", "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?", "commentary": "A conjecture of Owings [Ow74]. Hindman [Hi79] has shown that this is false for $3$-colourings. \n\nIf we do not ask for the elements $2a$ for $a\\in A$ to also be the same colour the answer is yes by Hindman's theorem (see [532]).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1199: [Er80,p.104]" }, { "number": 1200, "url": "https://www.erdosproblems.com/1200", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots0$ there exists a set of primes $P$ such that $\\sum_{p\\in P}\\frac{1}{p}\\leq C$ and the number of integers $n\\leq x$ divisible by at least one $p\\in P$ is $\\gg_C x$.\n\nSee also [783] and [784].\n\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1200: [Er80,p.106]" }, { "number": 1201, "url": "https://www.erdosproblems.com/1201", "status": "open", "prize": "no", "tags": [ "number theory", "primes" ], "oeis": [], "formalized": "no", "statement": "Is it true that for every $\\epsilon,\\eta>0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?", "commentary": "Erdős wrote he could prove this for $\\epsilon=1/2$.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n View history", "references": "#1201: [Er80,p.107]" }, { "number": 1203, "url": "https://www.erdosproblems.com/1203", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\\[F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.\\]Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.", "commentary": "It is easy to prove that $F(n)\\geq 1-o(1)$. \n\nSee also [248], [679], and [890]\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#1203: [Er80,p.107]" }, { "number": 1204, "url": "https://www.erdosproblems.com/1204", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "We call a sequence of integers $0\\leq a_1<\\cdots k$. The lower order terms have been improved by Hensley and Richard [HeRi73] (see Section 10 of [Po14c] for more details).\n\nThe lower bound is originally due to Elliott [El65], but was rediscovered by the Polymath project on bounded gaps between primes (see [Po14c]).\n\nAn upper bound of $\\pi(x+y)\\leq \\pi(x)+(1+o(1))\\pi(y)$ (see [855]) together with the prime tuples conjecture would imply $A(k)\\geq (1+o(1))k\\log k$.\n\nIt is trivial that $B(k)k$ implies\\[B(k)\\leq (1/2+o(1))k\\log k,\\]and since $a_j\\geq A(j)$ we have $B(k) \\geq \\frac{1}{k}\\sum_{1\\leq j\\leq k}A(j)$, whence if $A(k)\\geq (c-o(1))k\\log k$ we have $B(k) \\geq (c/2-o(1))k\\log k$, and so it is likely that\\[B(k)\\sim(1/2+o(1))k\\log k.\\]In [Er80] Erdős also asks about the behaviour of $a_k$ where $a_1a_{i-1}$ such that $a_1,\\ldots,a_i$ is admissible).\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 07 April 2026. View history", "references": "#1204: [Er80,p.108]" }, { "number": 1206, "url": "https://www.erdosproblems.com/1206", "status": "open", "prize": "no", "tags": [ "number theory", "sidon sets" ], "oeis": [], "formalized": "no", "statement": "Does $\\{1,2^3,\\ldots,N^3\\}$ contain a Sidon set of size $\\gg N$?Is there an infinite set $A\\subset \\mathbb{N}$ of positive density such that $\\{a^3 : a\\in A\\}$ is a Sidon set?", "commentary": "This question may also be asked for higher powers, and [324] suggests that for fifth powers $\\{n^5 :n\\geq 1\\}$ may itself be a Sidon set.\n\nGabdullin and Konyagin [GaKo24] proved that there is a constant $c>0$ such that $\\{ n^3 : N-cN^{1/2}\\leq n\\leq N\\}$ is a Sidon set. Garaev, Garayev, and Konyagin [GGK26] have proved that the exponent $1/2$ can be improved to $4/7-o(1)$ for infinitely many $N$, and to $3/5$ for all $N$ if we replace $n^3$ with $n^4$.\n\nIn [Er80] Erdős defines $g_k(A)$ to be the maximal size of a Sidon set of $\\{a^k : a\\in A\\}$, and asks whether $g_k(A)\\geq g_k(\\{1,\\ldots,N\\})$ where $N=\\lvert A\\rvert$. (See [530] for the linear analogue.)\n\nSee also [773] for an analogous question about squares.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1206: [Er80,p.109]" }, { "number": 1207, "url": "https://www.erdosproblems.com/1207", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "Let $P_d(n)$ be such that in any set of $n$ points in $\\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle. Estimate $P_d(n)$ - in particular, is it true that\\[P_2(n)0$?", "commentary": "Erdős [Er80] attributes this problem to Riddell, and states it is not hard to prove that $P_d(n)>n^{\\epsilon_d}$ where $\\epsilon_d\\to 0$ as $d\\to \\infty$. Indeed, one can take any subset of points with all distinct distances, whence\\[P_d(n)\\geq f_d(n) \\geq n^{\\frac{1}{3d-3}-o(1)}\\]with $f_d(n)$ defined as in [1208].\n\nErdős suggests that perhaps the worst set of points is the set of lattice points in a sphere of suitable radius.\n\nThe case $d=1$ is equivalent to finding sets with no three-term arithmetic progressions, and is the $k=3$ case of [201]. \n\nThe bound of Pach and Tardos [PaTa02] on the maximum number of isosceles triangles in a point set, together with a random deletion argument, shows that $P_2(n) \\gg n^{0.432}$. In Section 5.3 of [BMP05] they claim that the regular polygon shows that $P_2(n) \\ll n^{1/2}$ but give no details. This seems to be incorrect, since an isosceles triangle from the vertices of a regular polygon correspond to three points whose angles form an arithmetic progression. This shows that $P_2(n)\\ll r_3(n)$, where $r_3(n)$ is the maximum size of a subset of $\\{1,\\ldots,n\\}$ without a non-trivial three-term arithmetic progression (see [140]).\n\nSee [1208] (and also [657]) for a variant in which we ask for all sets of three points to have distinct distances.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1207: [Er80,p.110]" }, { "number": 1208, "url": "https://www.erdosproblems.com/1208", "status": "open", "prize": "no", "tags": [ "geometry", "distances" ], "oeis": [], "formalized": "no", "statement": "For $d\\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\\to \\infty$.", "commentary": "It is known that\\[\\frac{n^{1/3}}{(\\log n)^{1/3}}\\ll F_2(n) \\ll \\frac{n^{1/2}}{(\\log n)^{1/4}}.\\]The upper bound is demonstrated by the set of integer lattice points in a square grid (which have $\\ll n/\\sqrt{\\log n}$ many distinct distances). The lower bound was proved by Charalambides [Ch13].\n\nThiele [Th95] proved $F_d(n) \\gg n^{\\frac{1}{3d-2}}$ for all $d\\geq 3$. This was improved to\\[F_d(n) \\gg_d n^{\\frac{1}{3d-3}}(\\log n)^{1/3-\\frac{2}{3d-3}}\\]by Conlon, Fox, Gasarch, Harris, Ulrich, and Zbarsky [CFGHUZ15]. The example of integer lattice points shows that $F_d(n)\\ll n^{1/d}$ for $d\\geq 2$.\n\nErdős and Guy [ErGu70] consider the problem of determining the maximal size of such a subset of the $n^{1/d}\\times\\cdots \\times n^{1/d}$ integer grid, showing that it is at least $n^{\\frac{2}{3d}-o(1)}$ (and at most $n^{1/d}$). Lefmann and Thiele [LeTh95] have improved the lower bound to $n^{\\frac{2}{3d}}$.\n\nThe case $d=1$ is the subject of [530], and Komlós, Sulyok, and Szemerédi [KSS75] have proved that $F_1(n)\\asymp n^{1/2}$.\n\nSee also [1207]. The behaviour of $F_d(n)$ for fixed $n\\geq 3$ as $d\\to \\infty$ is the subject of [1088].\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1208: [Er57b][Er80,p.110]" }, { "number": 1209, "url": "https://www.erdosproblems.com/1209", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $A=\\{a_1a_{k-1}$ be a prime such that $a_k+k\\equiv \\pmod{q_k}$, where $q_k$ is some prime not dividing $k$. This sequence can be made to grow arbitrarily fast. A similar construction with $\\pmod{q_k^2}$ provides a counterexample to the squarefree question.\n\nSee also [429] and [1102]. \n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1209: [Er80,p.111]" }, { "number": 1210, "url": "https://www.erdosproblems.com/1210", "status": "open", "prize": "no", "tags": [ "number theory" ], "oeis": [], "formalized": "no", "statement": "Let $A\\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\\in A$. Is it true that\\[\\sum_{a\\in A}\\frac{1}{n-a}\\leq \\sum_{p1$ and at least one of $x$ or $y$ is composite?", "commentary": "Herzog and Stewart studied the graph $G$, which they called visible lattice points, and proved that there is only one infinite component, and conjectured that if $p$ is prime and $p\\nmid a$ then $(a,p)$ always belongs to the infinite component. (This is according to [Er80]; I can find no such results of Herzog and Stewart, who have a paper [HeSt71] on which finite patterns appear in the set of visible lattice points, but does not address this question.)\n\nErdős originally asked this question with just the condition that $\\min(x,y)>1$ and (as he recounts in [Er80]) 'foolishly offered 25 dollars for a proof. In the evening Stewart gave the simple proof: $(p_k,p_{k+1})$ can be joined to $(p_{k+1},p_{k+2})$'. Indeed, if $p_k$ is the $k$th prime then there is a path\\[(p_k,p_{k+1})\\to (p_k,p_{k+1}+1)\\to \\cdots \\]\\[\\to (p_{k},p_{k+2})\\to (p_{k}+1,p_{k+2})\\to (p_{k+1},p_{k+2}).\\]This is permissible provided $[p_{k+1},p_{k+2}]$ contains no multiple of $p_k$, which is true provided $p_{k+2}<2p_{k}$, which is true for all $k\\geq 4$.\n\nOne could further demand that the path be monotone (i.e. every step increases the distance from the origin). Is there a monotone path where we change direction after a bounded number of steps?\n\nThe first $50\\times 50$ block of the graph $G$ restricted to those vertices $(x,y)$ with both $x,y>1$ and at least one of $x$ or $y$ composite is shown here.\n \n \n \n\n\n View the LaTeX source\n\n\n\n\n \n This page was last edited 08 April 2026. View history", "references": "#1212: [Er80,p.114]" } ]