The product conjecture for finite transitive sets in Euclidean Ramsey theory

Let $X\subset\mathbb{R}^m$ be a finite transitive set. A set $Y\subset\mathbb{R}^d$ is $k$-Ramsey for $X$ if every $k$-colouring of $Y$ contains a subset congruent to $X$. For a positive integer $n$, regard $X^n$ as a subset of $\mathbb{R}^{mn}$, and call $\lambda X^n=\{\lambda x:x\in X^n\}$ a scaling of $X^n$.

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 $X$.

This strengthens the usual approach to proving that sets are Ramsey, in which a finite transitive set is embedded and then a large product is coloured. The conjecture is presented as the geometric statement later reformulated in purely combinatorial terms; its resolution is not specified in the supplied text.