Approximations to m-Colored Complete Infinite Hypergraphs

Given an edge colouring of a graph with a set of m colours, we say that the graph is (exactly) m-coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number m > 2 and for all sufficiently large k, there is a k-colouring of the complete graph on N such that no complete infinite subgraph is exactly m-coloured. In the light of this result, we consider the question of how close we can come to finding an exactly m-coloured complete infinite subgraph. We show that for a natural number m and any finite colouring of the edges of the complete graph on N with m or more colours, there is an exactly m̂-coloured complete infinite subgraph for some m̂ satisfying |m− m̂| ≤ √ m/2 + 1/2; this is best-possible up to the additive constant. We also obtain analogous results for this problem in the setting of r-uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalisations of this conjecture to r-uniform hypergraphs.