  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.
