  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. In 1999, Stacey
and Weidl, partially resolving a conjecture of Erickson from 1994, showed that
for a fixed natural number $m>2$ and for all sufficiently large $k$, there is a
$k$-colouring of the complete graph on $\mathbb{N}$ such that no complete
infinite subgraph is exactly $m$-coloured. In the light of this result, we
consider the question of how close we can come to finding an exactly
$m$-coloured complete infinite subgraph. We show that for a natural number $m$
and any finite colouring of the edges of the complete graph on $\mathbb{N}$
with $m$ or more colours, there is an exactly ${\hat m}$-coloured complete
infinite subgraph for some ${\hat m}$ satisfying $|m-{\hat m}|\le \sqrt{m/2} +
1/2$; this is best-possible up to the additive constant. We also obtain
analogous results for this problem in the setting of $r$-uniform hypergraphs.
Along the way, we also prove a recent conjecture of the second author and
investigate generalisations of this conjecture to $r$-uniform hypergraphs.
