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, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's 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.

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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, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's 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.M S C 2 0 2 0 05C57 (primary), 05C15, 05C63 (secondary) INTRODUCTIONGames have been of interest to mathematicians for centuries. The field was sparked by the analysis of historic games such as tic-tac-toe. This is a finite game and naturally, this was the first class of games analysed. In recent decades, infinite games have increasingly drawn the attention of researchers. An intuitive starting point is just moving the rules of a finite game to an infinite board.Consider the aforementioned tic-tac-toe. Its counterpart on an infinite board became known as unrestricted 3-in-a-row and was further generalised to 𝑛-in-a-row (Beck [1]). As some results in the finite version simply rely on the fact that there are only finitely many possible plays, this

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