800 problems

An old problem of R.L. Graham and myself states: Is it true that if $m_k$ is sufficiently large and we colour the integers $2 \leq t \leq m_k$ by $k$ colours then … is always solva…

One last Ramsey type problem: Let $n_k$ be the smallest integer (if it exists) for which if we colour the proper divisors of $n_k$ by $k$ colours then $n_k$ will be a monochromatic…

Graham's conjecture. For every triangle $T$,

Let $T$ be an oriented tree on $n$ vertices, and let $\vv{r}(T)$ be the least $N$ such that every tournament on $N$ vertices contains a copy of $T$. Sumner's conjecture. … This is…

Let $K_n$ be a complete graph whose edges are coloured with two colours. A monochromatic cycle is a cycle all of whose edges have the same colour. Lehel's conjecture. The vertex se…

Erdős–Szekeres conjecture.

Fox–Grinshpun–Pach conjecture.

A Paley graph is the Cayley graph associated with the quadratic residues in a finite field of odd order. A graph is Ramsey when both its clique number and independence number are…

Ge, Xu, and Zhang's conjecture. For every fixed $t$,

Let a finite partition of $\mathbb{N}$ be given. Hindman's question. There exist $x,y\in\mathbb{N}$ such that … is monochromatic. This asks whether every finite coloring of the pos…

Kohayakawa–Kreuter Conjecture for Families. For tuples $(\mathcal{H}_1,\ldots,\mathcal{H}_r)$ that avoid the pathological cases, a threshold for…

Rado's conjecture. For every $n\in\mathbb{Z}^+$, there exists a linear homogeneous equation over $\mathbb{Z}$ with degree of regularity equal to $n$.

Let $C_n$ be the cycle on $n$ vertices and $K_k$ the complete graph on $k$ vertices, with $n\geq k\geq 3$. Erdős–Faudree–Rousseau–Schelp conjecture. The Ramsey number satisfies … T…

Let $q\geqslant 2$ and let $\mathrm{KG}^q(n,q)$ be the $q$-uniform Kneser hypergraph on the $q$-subsets of $[n]$, with edges consisting of $q$ pairwise disjoint vertices. Let…

Square-free grid subgraph conjecture. The grid subgraph $H$ lies in $\mathbf{PGR}$ if and only if $H$ is square-free.

Let $k\geq 2$ and let $n\geq 3$ be odd. Write $R_k(C_n)$ for the $k$-color Ramsey number of the cycle $C_n$, namely, the least integer $N$ such that every $k$-edge-coloring of…

Allen–Brightwell–Skokan conjecture. Equality should hold whenever $n\geq |H|chi(H)$.

Let $\mathcal{P}^k_n$ and $\mathcal{C}^k_n$ denote the $k$-uniform loose path and loose cycle with $n$ edges, respectively, and let $R(H_1,H_2)$ be the two-colour Ramsey number for…

Collapse conjecture.

Erdős–Faudree–Sós conjecture. Every $n$-vertex graph $G$ with $\text{hom}(G)=O(\log n)$ satisfies

Let $G$ be a non-empty graph, and let $F$ be the graph obtained from $G$ by deleting a vertex. Write $R(G)$ and $R(F)$ for their classical two-color Ramsey numbers. Conlon–Fox–Suda…

For a fixed $k\geq 2$, let $G^k_{n,p}$ be the random $k$-uniform hypergraph, and let $H$ be a fixed $k$-graph. The $k$-graph analogue replaces graphs by $k$-graphs, $G_{n,p}$ by…

Feder–Subi conjecture. For every $d \in \mathbb{N}$, every $2$-edge-colouring of $Q_d$ contains vertices $u,v$ with $\operatorname{dist}(u,v)=d$ that are connected by a path with a…

Let $P_k$ denote the path on $k$ vertices, and let $\tilde{r}(P_k,P_\ell)$ be the two-color online Ramsey number for the pair of paths $P_k$ and $P_\ell$. Cyman–Dzido–Lapinskas–Lo…

Let $\delta>0$. Consider the collection of non-oriented graphs on the vertex set $[n]$, with density measured relative to the number of such graphs. Gowers' clique-difference conje…