On the chromatic number of a random hypergraph

TL;DR

This paper generalizes the understanding of the chromatic number in random hypergraphs, extending previous results from graphs to hypergraphs with uniform edges, providing a complete characterization as the number of vertices grows large.

Contribution

It extends the known two-value characterization of the chromatic number from random graphs to the broader class of random uniform hypergraphs.

Findings

Chromatic number of random hypergraphs has a predictable limiting behavior.

The result generalizes the two-value phenomenon from graphs to hypergraphs.

Provides a complete characterization as the number of vertices tends to infinity.

Abstract

We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.