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arXiv · 2309.05526

The Rational Number Game

Nathan Bowler·Florian Gut
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Abstract

We investigate a game played between two players, Maker and Breaker, on a countably infinite complete graph where the vertices are the rational numbers. The players alternately claim unclaimed edges. It is Maker's goal to have after countably many turns a complete infinite graph contained in her coloured edges where the vertex set of the subgraph is order-isomorphic to the rationals. It is Breaker's goal to prevent Maker from achieving this. We prove that there is a winning strategy for Maker in this game. We also prove that there is a winning strategy for Breaker in the game where Maker must additionally make the vertex set of her complete graph dense in the rational numbers.

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Nathan Bowler, Florian Gut. 2023-09-11. The Rational Number Game. https://doi.org/10.37236/12380

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