MAKER–BREAKER GAMES ON AND

Abstract We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy the host graph. establish firm dependence outcome game axiomatic framework. Relating this, we prove that there winning strategy for Maker $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and Breaker ZFC+CH. similar result -game. Here, has ZF+DC+AD, while one ZFC+CH again.