Conjectured by Faber, Lovász, and Erdős. Kahn [Ka92] proved that $\chi(G)\leq (1+o(1))n$. Hindman has proved the conjecture for $n<10$. Kang, Kelly, Kühn, Methuku, and Osthus [KKKMO21] have proved the answer is yes for all sufficiently large $n$.

Conjectured by Bollobás and Erdős [BoEr76], who proved the existence of such a graph with $(1/8+o(1))n^2$ many edges. Solved by Fox, Loh, and Zhao [FLZ15], who proved that for every $n\geq 1$ there exists a graph on $n$ vertices with $\geq n^2/8$ many edges, containing no $K_4$, whose largest independent set has size at most \[ \ll \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}}n.\]

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.

Győri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].

Conjectured by Hajnal and Erdős and solved by Liu and Montgomery [LiMo20]. The lower density of the set can be $0$ since there are graphs of arbitrarily large chromatic number and girth.

Conjectured by Bollobás and Erdős. Bollobás and Shelah have confirmed this for $k=1$. Proved by Gyárfás [Gy92], who proved the stronger result that, if $G$ is 2-connected, then $G$ is either $K_{2k+2}$ or contains a vertex of degree at most $2k$.

A stronger form was established by Gao, Huo, and Ma [GaHuMa21], who proved that if a graph $G$ has chromatic number $\chi(G)\geq 2k+3$ then $G$ contains cycles of $k+1$ consecutive odd lengths.

If $G$ is not bipartite the answer is yes, proved by Erdős, Frankl, and Rödl [ErFrRo86]. The answer is no for $G=C_6$, the cycle on 6 vertices. Morris and Saxton [MoSa16] have proved there are at least \[2^{(1+c)\mathrm{ex}(n;C_6)}\] such graphs for infinitely many $n$, for some constant $c>0$. It is still possible (and conjectured by Morris and Saxton) that the weaker bound of \[2^{O(\mathrm{ex}(n;G))}\] holds for all $G$.

Conjectured by Erdős and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.

He, Ma, and Yang [HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.

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 \[\geq \exp(c_H\sqrt{\log n\log\log n}).\]

Erdős, Hajnal, and Shelah have shown that $G_1$ and $G_2$ must both contain all sufficiently large cycles.

Conjectured by Mihók and Erdős. It is likely that $2^n$ can be replaced by any sufficiently quickly growing sequence (e.g. the squares).

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 $k\geq 2$.

This 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 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 $m$.

Gyárfás, Komlós, and Szemerédi [GyKoSz84] have proved that this sum is $\gg \log k$. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.

This has been proved by Bollobás [Bo77]. The best dependence of the constant $c(a,b)$ is unknown.

Bollobás proved that such a $c$ does exist if $A$ is an infinite arithmetic progression containing even numbers. Erdős was 'almost certain' that if $A$ is the set of powers of $2$ then no such $c$ exists (although conjectures that $n$ vertices and average degree $\gg (\log n)^{C}$ suffices for some $C=O(1)$). If $A$ is the set of squares (or the set of $p\pm 1$ for $p$ prime) then he had no guess.

Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set $A$ exists.

Liu and Montgomery [LiMo20] proved that in fact this is true when $A$ is the set of powers of $2$ (more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős.

Proved by Reed [Re99]. (Thanks also to Reed for pointing out that the case $k=0$ is trivial, since if $G$ is not bipartite then $G$ contains an odd cycle.)

Conjectured by Erdős, Hajnal, and Szemerédi [ErHaSz82]. Rödl [Ro82] has proved this for hypergraphs. It 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 $\aleph_1$.

Conjectured by Erdős, Hajnal, and Szemerédi [ErHaSz82].

Conjectured by Erdős, Faudreee, and Ordman. This would be best possible, as witnessed by dividing the vertices of $K_n$ into two equal parts and colouring all edges between the parts red and all edges inside the parts blue.

The answer is yes, proved by Gruslys and Letzter [GrLe20].

Erdős offers \$100 for just a proof of the existence of this constant, without determining its value. He also offers \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. 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 [CaGrMoSa23].

Erdős gave a simple probabilistic proof that $R(k) \gg k2^{k/2}$. Equivalently, 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 $c>0$. 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 $C>0$.

Asked by Erdős, Faudree, Rousseau, and Schelp [ErFaRoSc93]. $K_4$ is the only known example of such a graph.

In other words, is $K_{3,3}$ Ramsey size linear? Asked by Erdős, Faudree, Rousseau, and Schelp [ErFaRoSc93].

Asked by Erdős, Ordman, and Zalcstein [ErOrZa93], who proved an upper bound of $(1/4-\epsilon)n^2$ many cliques (for some very small $\epsilon>0$).

Conjectured by Erdős, Fajtlowicz, and Staton. It is known that $F(5)=3$ and $F(7)=4$. Ramsey'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 $n^{1/2}(\log n)^{O(1)}$.

Conjectured by Erdős and Faudree, who showed that $f(n) > 2^{n/2}$, and further speculate that $f(n)/2^{n/2}\to \infty$. This conjecture was proved by Verstraëte [Ve04], who proved the number of such sets is \[\ll 2^{n-n^c}\] for some constant $c>0$.

A weaker version of the conjecture asks for some constant $c$ such that $f(m)>f(n)-c$ for all $m>n$. This question can be asked for other graphs than $C_4$.

The best known result is due to Balogh, Hu, Lidicky, and Liu [BHLL14], who proved that $0.6068 n2^{n-1}$ edges suffice.

Erdős originally conjectured that $R(G)\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$.

Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 1/4$. 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 $R(k)$.

Conjectured by Erdős and McKay, who proved it with $\delta n^2$ replaced by $\delta (\log n)^2$. Solved by Kwan, Sah, Sauermann, and Sawhney [KSSS22]. Erdős' original formulation also had the condition that $G$ has $\gg n^2$ edges, but an old result of Erdős and Szemerédi says that this follows from the other condition anyway.

Conjectured by Erdős and Hajnal. Rödl [Ro77] has proved the $k=3$ case. The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.

Conjectured by Erdős, Hajnal, and Szemerédi [ErHaSz82]. This fails if the graph has chromatic number $\aleph_0$.

Conjectured by Erdős, Hajnal, and Szemerédi [ErHaSz82].

Conjectured by Erdős and Simonovits. Disproved by Janzer [Ja21].

Conjectured by Erdős, Kohayakava, and Gyárfás. Edwards [Ed73] proved that $f(m)\geq 0$ always. Note that $f(\binom{n}{2})= 0$, taking $K_n$. Solved by Alon [Al96], who showed $f(n^2/2)\gg n^{1/2}$, and also showed that $f(m)\ll m^{1/4}$ for all $m$. The best possible constant in $f(m)\leq Cm^{1/4}$ is unknown.

A blown-up $C_5$ (or, as Simonovits observed, a blown-up Petersen graph) shows that this would be best possible.

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. It is 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: the 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), as soon as $N \geq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \geq C^{n}$, contradicting the conjecture. Perhaps 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.

Conjectured by Andrásfai and Erdős. It is possible that such a graph could contain an infinite complete graph.

Asked by Erdős and Pach. Simonovits observed that the subsets of $[3m-1]$ of size $m$, two sets joined by edge if and only if they are disjoint, forms a triangle-free graph of diameter $2$ which is regular of degree $\binom{2m-1}{m}$, showing that $f(n)=o(n)$ infinitely often. This graph may have the minimal possible $f$, but Erdős encourages the reader to try and find a better graph.

Asked by Erdős and Gyárfás, who proved that this is the case when $G$ has maximum degree $\ll \log n/\log\log n$. A construction of Simonovits shows that this conjecture is false if we just have maximum degree $\leq Cn^{1/2}$, for some large enough $C$.

Asked by Erdős and Gyárfás, who proved that \[\frac{5}{6}(n-1) < f(n)<n,\] and that $f(9)=8$. Erdős believed the upper bound is closer to the truth. In fact the lower bound is: Bennett, Cushman, Dudek,and Pralat [BCDP22] have shown that \[f(n) \sim \frac{5}{6}n.\] Joos and Mubayi [JoMu22] have found a shorter proof of this.

Conjectured by Erdős and Simonovits. Open even for $r=3$.

Conjectured by Erdős and Simonovits. Disproved by Janzer [Ja21].

Asked by Erdős and Nešetřil. They 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 independently by Chung-Trotter and Gyárfás-Tuza.

Asked by Erdős and Nešetřil. Seymour observed that $c(3m+2)\geq 3^m$, as seen by the graph of $m$ independent paths of length $4$ joining two vertices.

It is easy to see that $h(n)\leq n-\sqrt{n}$ and that $h(n)\leq n-H(n)$. Conjectured by Erdős and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. Erdős remarked that this conjecture is 'perhaps completely wrongheaded'.

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 [ErFaRoSc78]), and the lower bound is due to Spencer [Sp77].

The Burr-Erdős conjecture. Solved by Lee [Le16], who proved that \[ R(H) \leq 2^{2^{O(d)}}n.\]

It is known that there exists some constant $c>0$ such that for large $k$ \[c\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}.\] 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 [AjKoSz80]. The lower bound has been improved to \[\left(\frac{1}{4}-o(1)\right)\frac{k^2}{\log k}\] 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.

Spencer [Sp77] proved \[R(4,k) \gg (k\log k)^{5/2}.\] Ajtai, Komlós, and Szemerédi [AjKoSz80] proved \[R(4,k) \ll \frac{k^3}{(\log k)^2}.\] This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that \[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

Erdős, Hajnal, and Rado proved that $R_r(k)$ grows like a tower of exponentials of height at least $r-2$ and at most $r-1$. Hajnal has proved that $r-1$ is the correct height if we consider the corresponding hypergraph Ramsey number for $4$ colours.

This 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.

Conjectured by Burr and Erdős. The 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.)

Asked by Erdős and Sauer. Resolved by Janzer and Sudakov [JaSu22], who proved that there exists some $C=C(k)>0$ such that any graph on $n$ vertices with at least $Cn\log\log n$ edges contains a $k$-regular subgraph.

A construction due to Pyber, Rödl, and Szemerédi [PRS95] shows that this is best possible.

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 bounds are all of the form $ck!+O(1)$, and arise from this 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 due to Exoo [Ex94], \[R(3;k) \gg (321)^{k/5}.\]

See also [483].

Conjectured by Erdős and Gallai. The 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.