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Given an edge colouring of a graph with a set of $m$ colours, we say that thegraph is (exactly) $m$-coloured if each of the colours is used. We consideredge colourings of the complete graph on $\mathbb{N}$ with infinitely manycolours and show that either one can find an $m$-coloured complete subgraph forevery natural number $m$ or there exists an infinite subset $X \subset\mathbb{N}$ coloured in one of two canonical ways: either the colouring isinjective 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 colourused on $X \setminus \lbrace v \rbrace$). This answers a question posed byStacey and Weidl in 1999. The techniques that we develop also enable us toresolve some further questions about finding $m$-coloured complete subgraphs incolourings with finitely many colours.

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