# Borsuk and Ramsey type questions in Euclidean space

## TL;DR

This paper surveys problems in diameter graphs and geometric Ramsey theory, and extends a theorem disproving Borsuk's conjecture to show high-dimensional point sets with specific coloring and distance properties.

## Contribution

It provides new results on coloring point sets in high dimensions to avoid monochromatic unit distances, extending known theorems in geometric combinatorics.

## Key findings

- Existence of high-dimensional point sets with controlled coloring properties
- Extension of Kahn and Kalai's theorem to multiple points and colorings
- Disproof of Borsuk's conjecture in certain high-dimensional contexts

## 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 $\varepsilon=\varepsilon(r)>0$ 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 $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.