532 problems

For every pair of integers $q\ge 2$ and $d\ge 1$, and every $d$-degenerate graph $H$, does Builder have a strategy in the $q$-color online Ramsey game that forces Painter to produc…

Let $F$ be an ordered forest on $n$ vertices, and let $\chi_{<}(F)$ denote its interval chromatic number. Is there a constant $C>0$ such that every such $F$ satisfies … Here…

For every pair of integers $s>k\ge 4$, there exists a constant $c=c(s,k)>0$ such that, for all sufficiently large integers $n$, $r_k(s,n)\ge \operatorname{twr}_{k-1}(cn)$. Here…

Determine whether there exists an absolute constant $c>0$ such that, for every integer $d\ge 1$, there is a constant $C_d>0$ for which every weakly $d$-degenerate ordered $3$-unifo…

For $n\ge 1$, let $T_n=\{(x,y,z)\in\{1,\ldots,n\}^3:x+y=z\}$ be the set of Schur triples, and for a coloring $c:\{1,\ldots,n\}\to\{1,2,3\}$ let $R_n(c)$ be the number of triples…

For every integer $k\ge 2$, every oriented graph $D$ with chromatic number $\chi(D)\ge 2k-2$ contains every oriented tree $T$ on $k$ vertices as an oriented subgraph. Equivalently,…

For an odd integer $m\ge 3$, determine the least integer $f(m)$ such that, for every integer $n\ge f(m)$ and every tree $T_n$ on $n$ vertices, the Ramsey number satisfies…

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A set $R\subseteq\mathbb{N}_{>0}$ is called a set of nice recurrence if, for every measure-preserving system $(X,\mu,T)$, every measurable set $A\subseteq X$, and every…

Determine the exact value of the Ramsey number $R(4,20)$, defined by…

For every $k,r\in\mathbb N$ with $k\ge2$ and $r\ge2$, does there exist a set $A\subseteq\mathbb N$ containing no non-trivial arithmetic progression of length $k+1$, such that every…

Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\cdots$ such that $a_n<f(n)$ for infinitely many $n$ and the set…

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 th…

The assertion is false: there exist a positive natural number $n$ and a coloring $\mathbb{R}^2\to\{1,\ldots,n\}$ such that no color class contains, for every real number $A>0$, poi…

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 tr…

For every natural number $k\geq 3$, does there exist a finite set $A\subset\mathbb R^2$ such that, for every coloring of $A$ with two colors, there is a subset $S\subseteq A$ satis…

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…

For any fixed $k\geq 3$, $R(k,n) \gg \frac{n^{k-1}}{(\log n)^c}$ for some constant $c=c(k)>0$.

Let $f(n)$ be the largest integer such that every tournament on $n$ vertices contains a transitive subtournament on $f(n)$ vertices. Is $f(n)=⌊\log_2 n⌋+1$?

Partition $N=A⊔B$, and let $S(A)$ and $S(B)$ be the sets of finite sums of distinct elements of $A$ and $B$. If $\overline δ_{\log}(X)=\limsup_{x→∞}(1/\log x)\sum_{n<x,n∈X}1/n$, how s…

In every 2-coloring $c:\mathbb N\to\{0,1\}$, does there exist an infinite set $A\subseteq\mathbb N$ such that all elements of its sumset … have the same color? Equivalently, must t…

In every two-coloring of the subsets of an $n$-element set, how large a monochromatic family must exist that is closed under both unions and intersections? How large if only closur…

Let $\hat r(G_1,G_2)$ be the least number of edges in a graph $H$ such that every red-blue colouring of $E(H)$ contains a red copy of $G_1$ or a blue copy of $G_2$. Let $f(n)$ be t…

For sets $A_1\subseteq A$, $B_1\subseteq B$, and $C_1\subseteq C$, a box $A_1\times B_1\times C_1$ is monochromatic under a coloring $f:A\to B\to C\to\operatorname{Fin}2$ if there…

The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in…