Maker-Breaker games on $ K_{\omega_1}$ and $K_{\omega,\omega_1}$

TL;DR

This paper studies Maker-Breaker positional games on uncountably large graphs, showing that the existence of winning strategies depends heavily on the underlying set-theoretic axioms.

Contribution

It establishes the dependence of Maker-Breaker game outcomes on different axiomatic frameworks, especially involving ZFC, CH, MA, and AD.

Findings

Maker wins under ZFC+MA+¬CH for K_{ω,ω₁}

Breaker wins under ZFC+CH for K_{ω,ω₁}

Maker wins under ZF+DC+AD for K_{ω₁}

Abstract

We investigate Maker-Breaker games on graphs of size $\aleph_1$ in which Maker's goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega,\omega_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega_1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.