Strong Ramsey Games: Drawing on an infinite board

Hefetz, Dan; Kusch, Christopher; Narins, Lothar; Pokrovskiy, Alexey; Requilé, Clément; Sarid, Amir

doi:10.48550/arxiv.1605.05443

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2020

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“…Although it might seem implausible that this could really happen, in fact for the analogous games played on 5-regular hypergraphs rather than graphs cases are known in which the second player can force a draw. Such an example was constructed by Hefetz, Kusch, Narins, Pokrovskiy, Requilé and Sarid in [5].…”

Section: Introductionmentioning

confidence: 99%

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

Bowler¹,

Emde²,

Gut³

2020

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%