A canonical Ramsey theorem for exactly $m$-coloured complete subgraphs

Abstract: 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. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an $m$-coloured complete subgraph for every natural number $m$ or there exists an infinite subset $X \subset \mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on $X$ or there exists a distinguished vertex $v$ in $X$ such that $X \setminus \lbrace v \rbrace$ is $1$-coloured and each edge between $v$ and $X \setminus \lbrace v \rbrace$ has a distinct colour (all different to the colour used on $X \setminus \lbrace v \rbrace$). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding $m$-coloured complete subgraphs in colourings with finitely many colours.