On rainbow-free colourings of uniform hypergraphs

TL;DR

This paper investigates the threshold for rainbow-free colourings in random k-uniform hypergraphs, establishing a precise probabilistic boundary for the existence of such colourings.

Contribution

It introduces the exact threshold function p* = (k-1)(ln n)/n for rainbow-free colourings in random hypergraphs, advancing understanding of hypergraph colourings.

Findings

p* is the threshold for rainbow-free colourings

Threshold depends on hypergraph uniformity k and size n

Provides probabilistic bounds for colouring existence

Abstract

We study rainbow-free colourings of $k$-uniform hypergraphs; that is, colourings that use $k$ colours but with the property that no hyperedge attains all colours. We show that $p^*=(k-1)(\ln n)/n$ is the threshold function for the existence of a rainbow-free colouring in a random $k$-uniform hypergraph.