A general framework for hypergraph colouring

TL;DR

This paper introduces a broad theorem for hypergraph colouring that improves bounds over traditional methods, demonstrating the abundance of colourings and applying to various colouring and combinatorial problems.

Contribution

It provides a new, general framework for hypergraph colouring that enhances existing bounds and simplifies proofs, inspired by recent advances in nonrepetitive colourings.

Findings

Matches or slightly improves Lovász Local Lemma bounds

Shows exponential number of colourings exist

Applies to multiple colouring and combinatorial problems

Abstract

The Lov\'asz Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for colouring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lov\'asz Local Lemma. Moreover, the theorem directly shows that there are exponentially many colourings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colourings by Rosenfeld [2020]. We apply our general theorem in the setting of proper hypergraph colouring, proper graph colouring, independent transversals, star colouring, nonrepetitive colouring, frugal colouring, Ramsey number lower bounds, and for $k$-SAT.