SOLVED

Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?

This follows from the colouring result of Croot [Cr03]. Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$ (simply taking $n_k$ to be the lowest common multiple of some interval $[1,C^k]$). Sawhney has observed that there is also a doubly exponential lower bound, and hence this bound is essentially sharp.

Indeed, we must trivially have $\sum_{d|n_k}1/d \geq k$, or else there is a greedy colouring as a counterexample. Since $\prod_{p}(1+1/p^2)$ is finite we must have $\prod_{p|n_k}(1+1/p)\gg k$. To achieve the minimal $\prod_{p|n_k}p$ we take the product of primes up to $T$ where $\prod_{p\leq T}(1+1/p)\gg k$; by Mertens theorems this implies $T\geq C^{k}$ for some constant $C>1$, and hence $n_k\geq \prod_{p\mid n_k}p\geq \exp(cC^k)$ for some $c>0$.

SOLVED - $100

A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$.

Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that \[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\] for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.\] Improve either of these bounds.

SOLVED - $250

A set of integers $A$ is Ramsey $r$-complete if, whenever $A$ is $r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. Prove any non-trivial bounds about the growth rate of such an $A$ for $r>2$.

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.

SOLVED

Is it true that in any $2$-colouring of the edges of $K_n$ there must exist at least
\[(1+o(1))\frac{n^2}{12}\]
many edge-disjoint monochromatic triangles?

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].

In [Er97d] Erdős also asks for a lower bound for the count of edge-disjoint monochromatic triangles in single colour (the colour chosen to maximise this quantity), and speculates that the answer is $\geq cn^2$ for some constant $c>1/24$.

OPEN - $250

Find the value of $\lim_{k\to \infty}R(k)^{1/k}$.

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

OPEN - $100

Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.

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 $C>0$.

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

SOLVED

We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.

Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are?

OPEN

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 $f_c(n)\gg \log n$?

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 $f_c(n)\to \infty$.

Edwards (unpublished) and Khadziivanov and Nikiforov [KhNi79] proved independently that $f_c(n) \geq n/6$ when $c>1/4$.

Fox 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.

The best known lower bounds for $f_c(n)$ are those from Szemerédi's regularity lemma, and as such remain very poor.

See also [600] and the entry in the graphs problem collection.

OPEN

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 $\chi(G)=k$?

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 $W$.

Since $R(k)\leq 4^k$ 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 $R(k)$.

This problem is #12 and #13 in Ramsey Theory in the graphs problem collection.

SOLVED - $100

For any $\epsilon>0$ there exists $\delta=\delta(\epsilon)>0$ such that if $G$ is a graph on $n$ vertices with no independent set or clique of size $\geq \epsilon\log n$ then $G$ contains an induced subgraph with $m$ edges for all $m\leq \delta n^2$.

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.

OPEN

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 directed path of size $m$. Determine $k(n,m)$.

SOLVED

Let $\alpha$ be a cardinal or ordinal number or an order type such that every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue $K_3$. For every $n\geq 3$ must every two-colouring of $K_\alpha$ contain either a red $K_\alpha$ or a blue $K_n$?

Conjectured by Erdős and Hajnal. In arrow notation, this is asking where $\alpha \to (\alpha,3)^2$ implies $\alpha \to (\alpha, n)^2$ for every finite $n$.

The answer is no, as independently shown by Schipperus [Sc99] (published in [Sc10]) and Darby [Da99].

For example, Larson [La00] has shown that this is false when $\alpha=\omega^{\omega^2}$ and $n=5$. There is more background and proof sketches in Chapter 2.9 of [HST10], by Hajnal and Larson.

OPEN

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
\[R(n;3,r) < C^{\sqrt{n}}.\]

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 $R(n;3,2) \geq C^{n}$:

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), provided $N \leq C^n$ 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 [EFRS78]), and the lower bound is due to Spencer [Sp77].

This problem is #17 in Ramsey Theory in the graphs problem collection.

OPEN - $500

Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the largest $m$ 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?

For $\alpha=0$ this is the usual Ramsey function. A 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 $1/2$.

Erdős believed there might be just one jump, occcurring at $\alpha=0$.

Conlon, 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 $\alpha=0$.

For all $\alpha>0$ it is known that \[F^{(t)}(n,\alpha)\gg_t (\log n)^{c_\alpha}.\] See also [563].

OPEN

Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that in any 2-colouring of the edges of $K_n$ any 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 $c_\alpha$.

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

SOLVED

For any $d\geq 1$ if $H$ is a graph such that every subgraph contains a vertex of degree at most $d$ then $R(H)\ll_d n$.

The Burr-Erdős conjecture. This is equivalent to showing that if $H$ is the union of $c$ forests then $R(H)\ll_c n$, and also that if every subgraph has average degree at most $d$ then $R(H)\ll_d n$. Solved by Lee [Le16], who proved that
\[ R(H) \leq 2^{2^{O(d)}}n.\]

This problem is #9 in Ramsey Theory in the graphs problem collection. See also [800].

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.

See also [544].

Spencer [Sp77] proved
\[R(4,k) \gg (k\log k)^{5/2}.\]
Ajtai, Komlós, and Szemerédi [AKS80] 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
\[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

This problem is #5 in Ramsey Theory in the graphs problem collection.

OPEN

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?

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 $\lvert A\rvert=2$.

OPEN

In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.

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.

OPEN

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 $\mathbb{R}^n$.

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].

OPEN

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that
\[R(Q_n) \ll 2^n.\]

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.)

OPEN - $250

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
\[\lim_{k\to \infty}R(3;k)^{1/k}.\]

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].

OPEN

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 $d$.

Originally asked by Cohen. Erdős 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). 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.

OPEN

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 $1$?

Juhász [Ju79] has shown that $k\geq 5$. Erdős and Graham claim that $k\leq 10000000$ ('more or less'), but give no proof.

Erdős 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.

It 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).

SOLVED

If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?

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.

SOLVED

Let $C>0$ be arbitrary. Is it true that, if $n$ is sufficiently large depending on $C$, 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
\[\sum_{x\in X}\frac{1}{\log x}\geq C?\]

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 \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\]

SOLVED

Is it true that, in any finite colouring of the integers, there must be two integers $x\neq y$ of the same colour such that $x+y$ is a square? What about a $k$th power?

A question of Roth, Erdős, Sárközy, and Sós [ESS89] (according to some reports, although in [Er80c] Erdős claims this arose in a conversation with Silverman in 1977). Erdős, Sárközy, and Sós [ESS89] proved this for $2$ or $3$ colours.

In other words, if $G$ is the infinite graph on $\mathbb{N}$ where we connect $m,n$ by an edge if and only if $n+m$ is a square, then is the chromatic number of $G$ equal to $\aleph_0$?

This is true, as proved by Khalfalah and Szemerédi [KhSz06], who in fact prove the general result with $x+y=z^2$ replaced by $x+y=f(z)$ for any non-constant $f(z)\in \mathbb{Z}[z]$ such that $2\mid f(z)$ for some $z\in \mathbb{Z}$.

See also [438].

SOLVED

Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class and $a\neq b$).

A conjecture of Roth.

Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that there are at least \[\frac{N}{2}-O(N^{1-1/2^{k+1}})\] many even numbers which are of this form. They also prove that if $k=2$ then there are at least \[\frac{N}{2}-O(\log N)\] many even numbers which are of this form, and that $O(\log N)$ is best possible, since there is a $2$-colouring such that no power of $2$ is representable as a monochromatic sum.

A refinement of this problem appears as Problem 25 on the open problems list of Ben Green.

OPEN

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?

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 $1$.

SOLVED

Is it true that, in any two-colouring of the edges of $K_n$, there exist $sqrt{n}$ monochromatic paths, all of the same colour, which cover all vertices?

A problem of Erdős and Gyárfás. Gerencsér and Gyárfás [GeGy67] proved that, if the paths do not need to be of the same colour, then two paths suffice. Erdős and Gyárfás [ErGy95] proved that $2\sqrt{n}$ vertices suffice, and observed that $\sqrt{n}$ would be best possible here.

Solved in the affirmative by Pokrovskiy, Versteegen, and Williams [PVW24].

OPEN

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 $F(k)$.

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.

Erdős 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 \[F(k) \geq 2^{2^{k-1}/k}.\]

SOLVED

If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums
\[ \sum_{n\in S}n\]
(as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?

OPEN

Show that
\[R(3,k+1)-R(3,k)\to\infty\]
as $k\to \infty$. Similarly, prove or disprove that
\[R(3,k+1)-R(3,k)=o(k).\]

This problem is #8 in Ramsey Theory in the graphs problem collection. See also [165].

OPEN

Show that if $G$ has $\binom{n}{2}$ edges then
\[R(G) \leq R(n).\]
More generally, if $G$ has $\binom{n}{2}+t$ edges with $t\leq n$ then
\[R(G)\leq R(H)\]
where $H$ is the graph formed by connected a new vertex to $t$ of the vertices of $K_n$.

In other words, are cliques extremal for Ramsey numbers. Asked by Erdős and Graham.

This problem is #10 in Ramsey Theory in the graphs problem collection.

This is true, and was proved by Sudakov [Su11]. The analogous question for $\geq 3$ colours is still open.

This problem is #11 in Ramsey Theory in the graphs problem collection.

Equality holds when $T$ is a star on $n$ vertices.

Implied by [548].

This problem is #14 in Ramsey Theory in the graphs problem collection.

SOLVED

If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then show that $R(T)=4k$.

It follows from results in [EFRS82] that $R(T)\geq 4k-1$.

This is false: Norin, Sun, and Zhao [NSZ16] have proved that if $T$ is the union of two stars on $k$ and $2k$ vertices, with an edge joining the centre of the two stars, then $R(T)\geq (4.2-o(1))k$. The best upper bound for the Ramsey number for this tree is $R(T)\leq 4.27492k+1$, obtained by Dubó and Stein [DuSt24].

This problem is #15 in Ramsey Theory in the graphs problem collection.

OPEN

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
\[R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.\]

Chvátal [Ch77] proved that $R(T,K_m)=(m-1)(n-1)+1$.

This problem is #16 in Ramsey Theory in the graphs problem collection.

SOLVED

Prove that
\[R(C_k,K_n)=(k-1)(n-1)+1\]
for $k\geq n\geq 3$ (except when $n=k=3$).

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 $R(C_k,K_n)$?

This identity was proved for $k>n^2-2$ by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to $k\geq 4n+2$.

Keevash, 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 $n$.

It was shown in [BEFRS89] that
\[n+\lceil\sqrt{n}\rceil+1\geq R(C_4,S_n)\geq n+\sqrt{n}-6n^{11/40}.\]
Füredi (unpublished) has shown that $R(C_4,S_n)=n+\lceil\sqrt{n}\rceil$ for infinitely many $n$.

SOLVED

Let $R(3,3,n)$ denote the smallest integer $m$ such that if we $3$-colour the edges of $K_m$ then there is either a monochromatic triangle in one of the first two colours or a monochromatic $K_n$ in the third colour. Define $R(3,n)$ similarly but with two colours. Show that
\[\frac{R(3,3,n)}{R(3,n)}\to \infty\]
as $n\to \infty$.

OPEN

Let $R(G;k)$ 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
\[\lim_{k\to \infty}\frac{R(C_{2n+1};k)}{R(K_3;k)}=0\]
for any $n\geq 2$.

A problem of Erdős and Graham. The problem is open even for $n=2$.

OPEN

Let $R(G;k)$ 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
\[R(C_{2n};k).\]

SOLVED

Let $R(G;3)$ 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
\[R(C_n;3) \leq 4n-3.\]

A problem of Bondy and Erdős. This inequality is best possible for odd $n$.

Luczak [Lu99] has shown that $R(C_n;3)\leq (4+o(1))n$ for all $n$, and in fact $R(C_n;3)\leq 3n+o(n)$ for even $n$.

Kohayakawa, 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 $R(C_n;3)=2n$.

OPEN

Let $R(G;k)$ 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
\[R(T;k)=kn+O(1)\]
for any tree $T$ on $n$ vertices?

A problem of Erdős and Graham. Implied by [548].

OPEN

Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine
\[R(K_{s,t};k)\]
where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.

Chung and Graham [ChGr75] prove the general bounds
\[(2\pi\sqrt{st})^{\frac{1}{s+t}}\left(\frac{s+t}{e^2}\right)k^{\frac{st-1}{s+t}}\leq R(K_{s,t};k)\leq (t-1)(k+k^{1/s})^s\]
and determined
\[R(K_{2,2},k)=(1+o(1))k^2.\]
Alon, Rónyai, and Szabó [ARS99] have proved that
\[R(K_{3,3},k)=(1+o(1))k^3\]
and that if $s\geq (t-1)!+1$ then
\[R(K_{s,t},k)\asymp k^t.\]

SOLVED

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 that is Ramsey for $G$.

If $G$ has $n$ vertices and maximum degree $d$ then prove that \[\hat{R}(G)\ll_d n.\]

#559:

A problem of Beck and Erdős. Beck [Be83b] proved this when $G$ is a path. Friedman and Pippenger [FrPi87] proved this when $G$ is a tree. Haxell, Kohayakawa, and Luczak [HKL95] proved this when $G$ is a cycle. An alternative proof when $G$ is a cycle (with better constants) was given by Javadi, Khoeini, Omidi, and Pokrovskiy [JKOP19].

This was disproved for $d=3$ by Rödl and Szemerédi [RoSz00], who constructed a graph on $n$ vertices with maximum degree $3$ such that \[\hat{R}(G)\gg n(\log n)^{c}\] for some absolute constant $c>0$. Tikhomirov [Ti22b] has improved this to \[\hat{R}(G)\gg n\exp(c\sqrt{\log n}).\] It is an interesting question how large $\hat{R}(G)$ can be if $G$ has maximum degree $3$. Kohayakawa, Rödl, Schacht, and Szemerédi [KRSS11] proved an upper bound of $\leq n^{5/3+o(1)}$ and Conlon, Nenadov, and Trujić [CNT22] proved $\ll n^{8/5}$. The best known upper bound of $\leq n^{3/2+o(1)}$ is due to Draganić and Petrova [DrPe22].

OPEN

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.

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].

Conlon, 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 $\hat{R}(K_{n,n})\asymp n^32^n$.

OPEN

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 $F_2=\cup_{j\leq t} K_{1,m_j}$. Prove that \[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.\]

Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS78] proved this when all the $n_i$ are identical and all the $m_i$ are identical.

OPEN

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 $r-1$?

OPEN

Let $F(n,\alpha)$ denote the largest $m$ 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 $0\leq \alpha\leq 1/2$, \[F(n,\alpha)\sim c_\alpha\log n\] for some constant $c_\alpha$ depending only on $\alpha$.

It is easy to show that, for every $0\leq \alpha\leq 1/2$,
\[F(n,\alpha)\asymp_\alpha \log n.\]

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.

See also [161].

OPEN - $500

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 \[R_3(n) \geq 2^{2^{cn}}?\]

OPEN

Let $R^*(G)$ be the induced Ramsey number: the minimal $m$ such that there is a graph $H$ on $m$ vertices such that any $2$-colouring of the edges of $H$ contains an induced monochromatic copy of $G$.

Is it true that \[R^*(G) \leq 2^{O(n)}\] for any graph $G$ on $n$ vertices?

A problem of Erdős and Rödl. Even the existence of $R^*(G)$ is not obvious, but was proved independently by Deuber [De75], Erdős, Hajnal, and Pósa [EHP75], and Rödl [Ro73].

Rödl [Ro73] proved this when $G$ is bipartite. Kohayakawa, Prömel, and Rödl [KPR98] have proved that \[R^*(G) < 2^{O(n(\log n)^2)}.\] An alternative (and more explicit) proof was given by Fox and Sudakov [FoSu08]. Conlon, Fox, and Sudakov [CFS12] have improved this to \[R^*(G) < 2^{O(n\log n)}.\]

OPEN

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
\[R(G,H)\ll m?\]

OPEN

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
\[R(G,H)\ll m?\]

In other words, is $G$ Ramsey size linear? A special case of [566]. In [Er95] Erdős specifically asks about the case $G=K_{3,3}$.

The 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.

OPEN

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,
\[R(G,H)\ll m?\]

In other words, is $G$ Ramsey size linear?

OPEN

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?

SOLVED - $100

Does there exist a graph $G$ with at most $10^{10}$ many vertices which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?

Erdős and Hajnal [ErHa67] first asked for the existence of any such graph. Existence was proved by Folkman [Fo70], but with very poor quantitative bounds. (As a result these quantities are often called Folkman numbers.) Let this particular Folkman number be denoted by $N$.

Frankl and Rödl [FrRo86] proved $N\leq 7\times 10^{11}$, which Spencer [Sp88] improved to $\leq 3\times 10^{9}$. This resolved the initial challenge of Erdős [Er75d] to beat $10^{10}$.

Lu [Lu07] proved $N\leq 9697$ vertices. The current record is due to Dudek and Rödl [DuRo08] who proved $N\leq 941$ vertices. For further information we refer to a paper of Radziszowski and Xu [RaXu07], who prove that $N\geq 19$ and speculate that $N\leq 127$.

SOLVED - $250

Let $\alpha$ be the infinite ordinal $\omega^\omega$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$?

A problem of Erdős and Rado. For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$.

This is true, and was proved by Chang [Ch72]. Milner modified his proof to prove that this remains true if we replace $K_3$ by $K_m$ for all finite $m<\omega$ (a shorter proof was found by Larson [La73]).

OPEN - $250

Let $\alpha$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$?

OPEN - $1000

Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$.

This property holds for $\beta=2$ and not for $3\leq \beta <\omega$ (Specker [Sp57]) and for $\beta=\omega$ (Chang [Ch72]).

The first open case is $\beta=\omega^2$ (see [591]). Galvin and Larson [GaLa74] have shown that if $\beta\geq 3$ has this property then $\beta$ must be 'additively indecomposable', so that in particular $\beta=\omega^\gamma$ for some $\gamma<\omega_1$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property.

See also [590].

OPEN

For which graphs $G_1,G_2$ is it true that

- 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
- for every graph $H$ without a $G_1$ there is an $\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.

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).

Whether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].

OPEN

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 $G$?

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 $\omega_1^2 \not\to (\omega_1\omega, K_{\aleph_0,\aleph_0})^2$.

SOLVED

Let $f(n)$ be minimal 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 $f(n)$. Does $f(n)\to \infty$ as $n\to \infty$?

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 $n$.

Day and Johnson [DaJo17] have shown that \[f(n)\geq 2^{c\sqrt{\log n}}\] for some constant $c>0$.

SOLVED

Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $\geq (1/8-c)n^2$ edges then $G$ must contain either a $K_4$ or an independent set on at least $n/\log n$ vertices?

A problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS93]. In other words, if $\mathrm{rt}(n;k,\ell)$ is the Ramsey-Turán number then is it true that
\[\mathrm{rt}(n; 4,n/\log n)< (1/8-c)n^2?\]
Erdős, Hajnal, Sós, and Szemerédi [EHSS83] proved that for any fixed $\epsilon>0$
\[\mathrm{rt}(n; 4,\epsilon n)< (1/8+o(1))n^2.\]
Sudakov [Su03] proved that
\[\mathrm{rt}(n; 4,ne^{-f(n)})=o(n^2)\]
whenever $f(n)/\sqrt{\log n}\to \infty$.

Resolved by Fox, Loh, and Zhao [FLZ15] who showed that the answer is no; in fact they prove that \[\mathrm{rt}(n; 4, ne^{-f(n)})\geq (1/8-o(1))n^2\] whenever $f(n) =o(\sqrt{\log n/\log\log n})$.

See also [22] and the entry in the graphs problem collection.

SOLVED

Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $\gg \log n$ many vertices. Must $G$ contain $\gg n^{5/2}$ induced subgraphs which pairwise differ in either the number of vertices or the number of edges?

A problem of Erdős, Faudree, and Sós, who proved there exist $\gg n^{3/2}$ many such subgraphs, and note that $n^{5/2}$ would be best possible.

This was proved by Kwan and Sudakov [KwSu21].

SOLVED

If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $\gg \log n$ vertices then $G$ contains an induced subgraph on $\gg n$ vertices which contains $\gg n^{1/2}$ distinct degrees.

A problem of Erdős, Faudree, and Sós.

This was proved by Bukh and Sudakov [BuSu07].

Jenssen, Keevash, Long, and Yepremyan [JKLY20] have proved that there must exist an induced subgraph which contains $\gg n^{2/3}$ distinct degrees (with no restriction on the number of vertices).

OPEN

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.

Erdős writes 'if the answer is affirmative many extensions and generalisations will be possible'.

SOLVED

If $\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?

Erdös writes 'perhaps this is easy or false'. It is not true for four-term arithmetic progressions: colour the integers in $[3^{2k},3^{2k+1})$ red and all others blue.

Ryan Alweiss has provided the following simple argument showing that the answer is yes: suppose we have some red/blue colouring without this property. Without loss of generality, suppose $1$ is coloured red, and then either $3$ or $5$ must be blue.

Suppose first that $3$ is blue. If $n\geq 6$ is red then (considering $1,n,2n-1$) we deduce $2n-1$ is blue, and then (considering $3,n+1,2n-1$) we deduce that $n+1$ is red. In particular the colouring must be eventually constant, and we are done.

Now suppose that $5$ is blue. Arguing similarly (considering $1,n,2n-1$ and $5,n+2,2n-1$) we deduce that if $n\geq 8$ is red then $n+2$ is also red, and we are similarly done, since the colouring must be eventually constant on some congruence class modulo $2$.

OPEN

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 $1\leq q\leq \binom{p-1}{2}+1$?

A problem of Erdős, Faudree, Rousseau, and Schelp.

When $q=1$ 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 $c(p,\binom{p-1}{2})\leq 1/2$.

SOLVED - $100

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 $G$.

Is it true that, if $P_n$ is the path of length $n$, then \[\hat{R}(P_n)/n\to \infty\] and \[\hat{R}(P_n)/n^2 \to 0?\]

A problem of Erdős, Faudree, Rousseau, and Schelp.

Answered by Beck [Be83b], who proved that in fact $\hat{R}(P_n)\ll n$.

SOLVED

Let $W(3,k)$ be the van der Waerden number defined as the minimum $n$ such that in any red/blue colouring of $\{1,\ldots,n\}$ there exists either a red $3$-term arithmetic progression or a blue $k$-term arithmetic progression.

Give reasonable bounds for $W(3,k)$. In particular, give any non-trivial lower bounds for $W(3,k)$ and prove that $W(3,k) < \exp(k^c)$ for some constant $c<1$.

While we do not have a full understanding of the growth of $W(3,k)$, both of the specific challenges of Erdős have been met.

Green [Gr22] established the superpolynomial lower bound \[W(3,k) \geq \exp\left( c\frac{(\log k)^{4/3}}{(\log\log k) ^{1/3}}\right)\] for some constant $c>0$ (in particular disproving a conjecture of Graham that $W(3,k)\ll k^2$). Hunter [Hu22] improved this to \[W(3,k) \geq \exp\left( c\frac{(\log k)^{2}}{\log\log k}\right).\] The first to show that $W(3,k) < \exp(k^c)$ for some $c<1$ was Schoen [Sc21]. The best upper bound currently known is \[W(3,k) \ll \exp\left( O((\log k)^9)\right),\] which follows from the best bounds known for sets without three-term arithmetic progressions (see [BlSi23] which improves slightly on the bounds due to Kelley and Meka [KeMe23]).

OPEN

Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\lfloor n^2/4\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges.

Is it true that \[F_k(n)\sim n^2/8?\]

A problem of Burr, Erdős, Graham, and Sós, who proved that
\[F_k(n)\gg n^2.\]

See also [810].

OPEN

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?

A problem of Burr, Erdős, Graham, and Sós.

See also [809].

OPEN

For which graphs $G$ does the following hold: for all large $n$ there exists some $d_G(n)$ 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 subgraph isomorphic to $G$ where each edge receives a different colour?

If $d_G(n)$ exists then determine the best possible value of $d_G(n)$.

A problem of Erdős, Pyber, and Tuza, who observe that if $d_G(n)$ exists then $d_G(n) < \frac{n-1}{e(G)}$.

The Kürschák competition in Hungary in 1986 asked students to prove that $d_{K_3}(n)$ exists. Kostochka proved that $d_{K_3}(n)=n/4$ is the best possible. Tuza proved that \[d_{C_4}(n) \leq \left(\frac{1}{4}-c\right)n\] for some constant $c>0$. Brightwell and Trotter proved that \[d_{C_6}(n) > (1-o(1))\frac{n}{6}.\]

OPEN

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
\[R(n+1)-R(n) \gg n^2?\]