Approximations to $m$-coloured 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 $\mathbb{N}$ 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 $\mathbb{N}$ with $m$ or more colours, there is an exactly ${\hat m}$-coloured complete infinite subgraph for some ${\hat m}$ satisfying $|m-{\hat m}|\le \sqrt{m/2} + 1/2$; 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.

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