420 problems

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…

Extremality conjecture. For all $t,r\geq 2$,

Polynomial bound conjecture. We have

Let $r(3,q)$ denote the two-color Ramsey number for a triangle and a clique of order $q$. For $q\geq 2$, the Ramsey–Turán density $\rho(3,2q-1)$ is defined using graphs admitting a…

Let $r$ and $t$ be positive integers, and let $r(t)$ denote the two-color Ramsey number: the least integer $m$ such that every $2$-coloring of the edges of the complete graph on…

Let $G$ be a closed oligomorphic permutation group, meaning a closed permutation group with finitely many orbits on $n$-tuples for every $n\geq 1$. An expansion of an $\omega$-cate…

The product conjecture. For every positive integer $k$, there exists a positive integer $n$ such that some scaling of $X^n$ is $k$-Ramsey for $X$.

Big-Line-Big-Clique Conjecture. For all integers $t$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane either contains $\ell$ co…

Let a coloring $\chi$ be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains a triangle $T$ if there is a monochromatic copy of $T$, where copies are obtained…

Let $T$ be a triangle in the Euclidean plane, and let a coloring be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains $T$ if it has a monochromatic copy of…

Let $\Delta(\varepsilon)$ denote the symmetric subset function studied in the source, with $0\leq\varepsilon\leq1$. Exact formula conjecture for $\Delta$. … for all…

Complete-graph 2-mean conjecture.

Mean Ramsey-function conjecture.

Let $r(C_p,K_r)$ denote the Ramsey number for a cycle $C_p$ versus a complete graph $K_r$, where $p$, $r$, and $k$ are positive integers. Nikiforov's asymptotic Ramsey conjecture.…

For $k\geq1$, let $\Delta_k(\varepsilon)$ denote the function studied in the paper, with $0\leq\varepsilon\leq1$. Delta three and Delta four polygonal conjecture. The graph of…

Ramsey-ordering conjecture. There exists such a Ramsey ordering for which, for all $\beta,\beta'\in\mathcal{B}$ and $\gamma,\gamma'\in\mathcal{C}$,

Let $K_n$ be the complete graph on $n$ vertices, let $\mathrm{Ramsey}(n,n)$ be the symmetric binary Ramsey number, and let $K_{\mathrm{Ramsey}(n,n)}$ be the complete graph on that…

Let $K_k$ and $K_n$ be complete graphs, let $E^r$ and $E^g$ be precolored red and green edge sets, and let $\mathrm{Ramsey}(n,n)$ denote the classic symmetric binary Ramsey number.…

Let $K_n$ be the complete graph on $n$ vertices, let $A$ be the achievement graph, and let $E^r$ and $E^g$ be the precolored red and green edge sets. Achievement-game tractability…

Let $G$ be a graph and let $A$ be the achievement graph, with no precolored red or green edges, denoted by $(G,A,\{\},\{\})$. Unrestricted graph Ramsey-game conjecture. Graph Ramse…

Let $H$ be a graph, and let $G$ be a graph of order $n$ with average degree $\overline d(G)$. Write $v(H)$ and $e(H)$ for the numbers of vertices and edges of $H$, and let…

Let $H$ be a graph, and let $G$ be a graph of order $n$. Write $v(H)$ and $e(H)$ for the numbers of vertices and edges of $H$, $\hom(H,G)$ for the number of homomorphisms from $H$…

Let $R(G,H)$ denote the two-colour Ramsey number, let $C_n$ be the cycle on $n$ vertices, and let $\chi(H)$ and $\sigma(H)$ denote the chromatic number and minimum colour-class size…

For graphs $G$ and $H$, let $R(G,H)$ denote the least integer $N$ such that every red-blue colouring of the edges of $K_N$ contains a red copy of $G$ or a blue copy of $H$. Erdős–F…