A note on diameter-Ramsey sets

TL;DR

This paper investigates diameter-Ramsey sets, showing that certain large circumradius sets, including triangles with angles over 135°, are not diameter-Ramsey, thus refining previous geometric Ramsey results.

Contribution

It establishes new geometric criteria for non-diameter-Ramsey sets, particularly relating to circumradius and angle size, improving prior bounds.

Findings

Sets with circumradius > 1/√2 are not diameter-Ramsey

Triangles with angles > 135° are not diameter-Ramsey

Some nearly regular simplices are not diameter-Ramsey

Abstract

A finite set $A \subset \mathbb{R}^d$ is called $\textit{diameter-Ramsey}$ 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 $A' \subset B$ 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 R\"odl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey.