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 N with exactly k colours, how small the set of natural numbers m for which there exists an exactly m-coloured complete infinite subgraph can be. We prove that this set must have size at least 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.