Exactly $m$-coloured complete infinite subgraphs

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. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson in 1994; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on $\mathbb{N}$ with exactly $k$ colours, how small the set of natural numbers $m$ for which there exists an $m$-coloured complete infinite subgraph can be. We prove that this set must have size at least $\sqrt{2k}$; this bound is tight for infinitely many values of $k$. We also obtain a version of this result for colourings that use infinitely many colours.

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