Borsuk and Ramsey type questions in Euclidean space

Abstract

We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\epsilon =\epsilon (r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset {\mathbb{R}}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\epsilon {)}^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.

second version addresses changes arising from the referee reports