  We investigate games played between Maker and Breaker on an infinite complete
graph whose vertices are coloured with colours from a given set, each colour
appearing infinitely often. The players alternately claim edges, Makers aim
being to claim all edges of a sufficiently colourful infinite complete subgraph
and Breakers aim being to prevent this. We show that if there are only finitely
many colours then Maker can obtain a complete subgraph in which all colours
appear infinitely often, but that Breaker can prevent this if there are
infinitely many colours. Even when there are infinitely many colours, we show
that Maker can obtain a complete subgraph in which infinitely many of the
colours each appear infinitely often.
