The $K^{\aleph_0}$ Game: Vertex Colouring

TL;DR

This paper studies an infinite vertex colouring game on complete graphs, showing how the number of colours influences Maker's ability to claim colourful infinite subgraphs, with results depending on whether the colours are finite or infinite.

Contribution

It introduces a new infinite graph game framework and characterizes Maker's winning strategies based on the finiteness of the colour set.

Findings

Maker can find a colourful infinite subgraph with finitely many colours.

Breaker can prevent a fully colourful infinite subgraph when colours are infinite.

Maker can still find infinitely many colours appearing infinitely often, even with infinitely many colours.

Abstract

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.