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'. 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].

A 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].

This problem is #3 in Ramsey Theory in the graphs problem collection.

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 $c>0$.

In [Er69b] Erdős asks for even a construction whose largest clique or independent set has size $o(n^{1/2})$, which is now known.

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 $C>0$.

This problem is #4 in Ramsey Theory in the graphs problem collection.

Asked by Erdős and Nešetřil in 1985 (see [FGST89]). This bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is sometimes called the strong chromatic index of $G$.

The weaker conjecture that there exists some $c>0$ such that $(2-c)\Delta^2$ sets suffice was proved by Molloy and Reed [MoRe97], who proved that $1.998\Delta^2$ sets suffice (for $\Delta$ sufficiently large). 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].

Erdős 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 independently by Chung-Trotter and Gyárfás-Tuza.

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

Solved by Bradač [Br24], who proved that $\alpha=\lim c(n)^{1/n}$ exists and \[\alpha \leq 2^{H(1/3)}=1.8899\cdots,\] where $H(\cdot)$ is the binary entropy function. Seymour's construction proves that $\alpha\geq 3^{1/3}=1.442\cdots$. Bradač conjectures that this lower bound is the true value of $\alpha$.

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'.

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.

A problem of Erdős and Nešetřil. Erdős [Er88] wrote 'This problem seems to be interesting only if there is a nice expression for $h_t(d)$.'

It is easy to see that $h_t(d)\leq 2d^t$ always and $h_1(d)=d+1$.

Erdős 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 $d$. This was proved by Chung, Gyárfás, Tuza, and Trotter [CGTT90].

Cambie, 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 $d\to \infty$).

The 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 \[h_t(d) \leq \tfrac{3}{2}d^t+1.\]