A note on diameter-Ramsey sets

Abstract

A finite set $A\subset {\mathbb{R}}^d$ is called $\text{diameter-Ramsey}$ if for every $r\in \mathbb{N}$, there exists some $n\in \mathbb{N}$ and a finite set $B\subset {\mathbb{R}}^n$ with $\mathrm{diam}(A)=\mathrm{diam}(B)$ such that whenever $B$ is coloured with $r$ colours, there is a monochromatic set $A^{\prime }\subset B$ which is congruent to $A$. We prove that sets of diameter $1$ with circumradius larger than $1/\sqrt{2}$ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than $135^{\circ }$ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and Rödl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey.

4 pages