This follows from the colouring result of Croot [Cr03]. Obtaining better quantitative aspects seems interesting still -- note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?

The answer is yes, as proved by Croot [Cr03].

The stated bounds are due to Burr and Erdős [BuEr85].

A paper of Burr and Erdős [BuEr85] proves both upper and lower bounds for $r=2$, showing that there exists some $c>0$ such that it cannot be true that \[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\] for all large $N$, and also constructing a Ramsey $2$-complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^3.\] Burr has shown that the sequence of $k$th powers is Ramsey $r$-complete for every $r,k\geq 1$.

Solved by Conlon, Fox, and Pham [CFP21], who constructed for every $r\geq 2$ an $r$-Ramsey complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,\] and showed that this is best possible, in that there exists some constant $c>0$ such that if $A\subset \mathbb{N}$ satisfies \[\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2\] for all large $N$ then $A$ cannot be $r$-Ramsey complete.

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 $\epsilon>0$).

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.

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.

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].

For $\alpha=0$ this is the usual Ramsey function. A conjecture of Erdős, Rado, and Hajnal implies that \[ F^{(t)}(n,0)\asymp \log_{t-1} n\] and results of Erdős and Spencer imply that \[F^{(t)}(n,\alpha) \asymp_\alpha (\log n)^{\frac{1}{t-1}}\] for $\alpha$ close to $1/2$.

It is easy to show with the probabilistic method that there exist $c_1(\alpha),c_2(\alpha)$ such that \[c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n.\]

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.

First asked by Hindman. Hindman [Hi80] has proved this is false (with 7 colours) if we ask for an infinite $A$.

Moreira [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.

Alweiss [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 $\lvert A\rvert=2$.

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.

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.

Sets known to be Ramsey include vertices of $k$-dimensional rectangles [EGMRSS73], non-degenerate simplices [FrRo90], trapezoids [Kr92], and regular polygons/polyhedra [Kr91].

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.)

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].

Originally asked by Cohen. Erdős observed that colouring according to whether $\{ \sqrt{2}n\}<1/2$ or not implies $f(d) \ll d$. Beck [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.

Juhász [Ju79] has shown that $k\geq 5$. It is also known that $k\leq 10000000$ (as Erdős writes, 'more or less').

Graham [Gr80] has shown that this is true if we replace rectangle by right-angled triangle. The same question can be asked for parallelograms. It is not true for rhombuses.

This is false; Kovač [Ko23] provides an explicit (and elegantly simple) colouring using 25 colours such that no colour class contains the vertices of a rectangle of area $1$. The question for parallelograms remains open.

The answer is yes, which was proved by Rödl [Ro03].

In the same article Rödl also proved a lower bound for this problem, constructing, for all $n$, a $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ such that if $X\subseteq \{2,\ldots,n\}$ is such that $\binom{X}{2}$ is monochromatic then \[\sum_{x\in X}\frac{1}{\log x}\ll \log\log\log n.\]

This bound is best possible, as proved by Conlon, Fox, and Sudakov [CFS13], who proved that, if $n$ is sufficiently large, then in any $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ there exists some $X\subset \{2,\ldots,n\}$ such that $\binom{X}{2}$ is monochromatic and \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\]

Erdős (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure.

Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms.

Schur proved that $f(k)<ek!$. See also [183].

A conjecture of Roth.