Nearly all known Euclidean Ramsey sets are subsoluble

Abstract

A finite set $X$ in a Euclidean space $\mathbb{R}^d$ is called Ramsey if for every $k$ there exists an integer $n$ such that whenever $\mathbb{R}^n$ is coloured with $k$ colours, there is a monochromatic copy of $X$. Graham conjectured that all spherical sets are Ramsey, but progress on this conjecture has been slow. A key result of K\v{r}\'{i}\v{z} is that all sets that embed in sets that are acted on transitively by a soluble group are Ramsey. We show that for nearly all known examples of Ramsey sets the converse is true, with only two possible exceptions.