Generalised Prisms and Euclidean Ramsey Theory

Abstract

A finite subset $X$ of $\mathbb R^d$ is called Ramsey if for every $k$ there exists an $n$ such that whenever $\mathbb R^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. K\v rí\v z showed that if there is a soluble group of symmetries of $X$ that acts transitively on $X$, then $X$ is Ramsey. Determining which sets are Ramsey is a major unsolved problem. In this paper we show that if there is a finite group of isometries of $\mathbb R^d$ that acts transitively on a set $X$, and also on a set $Y$, then the `prism' formed by $X$ and $Y$ in $\mathbb R^{d+1}$ (meaning the set $X$ together with a translate of $Y$ in the direction perpendicular to $\mathbb R^d$) is itself contained in a finite set on which a group of isometries acts transitively. Moreover, if the initial group of isometries is soluble then so is the final group. This provides a new tool for generating Ramsey sets.