On rainbow-free colourings of uniform hypergraphs

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 $p^{\ast }=(k-1)(\ln n)/n$ is the threshold function for the existence of a rainbow-free colouring in a random $k$-uniform hypergraph.