22 problems

Leader–Russell–Walters' conjecture. A finite set $X$ is Ramsey if and only if it is a subset of some finite transitive set $Y$.

A finite set $X\subset\mathbb{R}^d$ is spherical if it lies on the surface of some $d$-dimensional sphere. A finite set $X$ is Ramsey if, for every $k$, there exists an integer $n$…

Let a finite set $X\subseteq\mathbb{R}^m$ be $2$-Ramsey if every $2$-coloring of $\mathbb{R}^m$ contains a monochromatic congruent copy of $X$. A triangle is non-equilateral if its…

Let $\ell$ and $d$ be positive integers. For an integer $n$, let $X=\{x_1,\ldots,x_\ell\}$ be any set of $\ell$ points on the unit sphere $\mathbb{S}^{n-1}$, and let…

One-point extension conjecture. The set $X\cup\{z\}$ is Ramsey.

Block sets conjecture. For every pair of positive integers $m$ and $k$ and every template $T$ over $[m]$, there exist positive integers $n$ and $d$ such that every $k$-colouring of…

Spherical Ramsey-set conjecture. A finite set $X$ is Ramsey if and only if it is spherical.

Conlon–Wu conjecture. For every non-spherical set $X$, there exists a natural number $m$ such that

Erdős et al.'s conjecture. For every non-equilateral three-point configuration $T$, every 2-coloring of $\mathbb{E}^2$ contains a monochromatic congruent copy of $T$ in one of the…

Let $T$ be a triangle in the Euclidean plane, and write $\mathbb R^2 \xrightarrow{2} T$ for the assertion that every $2$-coloring of $\mathbb R^2$ contains a monochromatic copy of…

Let $T$ be a triangle, and let $\chi(\mathbb{R}^2,T)$ be the minimum number of colors needed to color the Euclidean plane so that no isometric copy of $T$ is monochromatic. Graham'…

Let $T$ be a triangle, and let $\chi(\mathbb{R}^2,T)$ be the minimum number of colors needed to color the Euclidean plane so that no isometric copy of $T$ is monochromatic. Erdős e…

Erdős's conjecture. No infinite set is universal in measure.

Let $X\subset\mathbb{E}^d$ be a set, and call it non-spherical if it is not contained in the surface of a sphere of any dimension. Write $\ell_m$ for the relevant $m$-point line co…

Let $X\subset\mathbb{E}^d$ be a set. Call $X$ Ramsey if for every natural number $r$ there exists $n$ such that every $r$-colouring of $\mathbb{E}^n$ contains a monochromatic copy…

For integers $n,k\geq 1$, let $\mathbb{E}^n$ denote $n$-dimensional Euclidean space, and let $\ell_s$ denote a set of $s$ equally spaced collinear points. Write…

Cyclic-quadrilateral conjecture. There exists a cyclic quadrilateral that is not subtransitive.

Uniform-template conjecture. For every pair of positive integers $m$ and $k$, and every template $\tau$ over $[m]$, there exist positive integers $n$ and $d$ such that every $k$-co…

Template conjecture. For every pair of positive integers $m$ and $k$, and every template $\tau$ over $[m]$, there exist positive integers $n$ and $d$ such that every $k$-colouring…

Block-permutation conjecture. For every pair of positive integers $m$ and $k$, there exist positive integers $n$ and $d$ such that every $k$-colouring of $[m]^n$ contains a block p…

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

The transitive-set conjecture. A finite set $X\subset\mathbb R^n$ is Ramsey if and only if it is congruent to a subset of a finite transitive set.