12 problems

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 $X$.

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…

A finite set $X\subset\mathbb{R}^d$ is Ramsey if, for every $k$, there exists an integer $n$ such that every $k$-colouring of $\mathbb{R}^n$ contains a monochromatic isometric copy…

A finite set $X$ in a Euclidean space is Ramsey if, for every $k\in\mathbb{N}$, some dimension contains a monochromatic congruent copy of $X$ under every $k$-colouring. Graham's bo…

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.

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 $X$.