# The chromatic number of random Borsuk graphs

## TL;DR

This paper investigates the chromatic number of a random geometric graph on a sphere, showing it remains at d+2 for small epsilon, with precise bounds on how epsilon can tend to zero, combining topological and probabilistic methods.

## Contribution

It establishes the asymptotic behavior of the chromatic number for random Borsuk graphs as epsilon approaches zero, providing optimal bounds and novel proof techniques.

## Key findings

- Chromatic number is d+2 for fixed small epsilon with high probability.
- The chromatic number remains d+2 if epsilon tends to zero slowly enough.
- If epsilon tends to zero faster, the graph is (d+1)-colorable with high probability.

## Abstract

We study a model of random graph where vertices are $n$ i.i.d. uniform random points on the unit sphere $S^d$ in $\mathbb{R}^{d+1}$, and a pair of vertices is connected if the Euclidean distance between them is at least $2- \epsilon$. We are interested in the chromatic number of this graph as $n$ tends to infinity. It is not too hard to see that if $\epsilon > 0$ is small and fixed, then the chromatic number is $d+2$ with high probability. We show that this holds even if $\epsilon \to 0$ slowly enough. We quantify the rate at which $\epsilon$ can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik--Schnirelman--Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor --- if $\epsilon \to 0$ faster than this, we show that the graph is $(d+1)$-colorable with high probability.