The Rational Number Game

TL;DR

This paper studies a strategic game played on the rational numbers' infinite graph, proving that both players have winning strategies under different conditions, revealing insights into infinite combinatorial game theory.

Contribution

It introduces and analyzes a new infinite graph game on rationals, establishing winning strategies for both Maker and Breaker under various constraints.

Findings

Maker has a winning strategy to build a complete subgraph isomorphic to the rationals.

Breaker has a winning strategy when Maker's subgraph must be dense.

The results advance understanding of infinite combinatorial games on ordered structures.

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.