论文标题
$ k_ {ω_1} $和$ k_ {ω，ω_1} $的制造商 - 破坏游戏
Maker-Breaker games on $ K_{ω_1}$ and $K_{ω,ω_1}$
论文作者
论文摘要
我们在尺寸$ \ aleph_1 $的图表上调查了Maker-Breaker游戏，其中制造商的目标是构建主机图的副本。我们在公理框架上建立了对游戏结果的坚定依赖。关于此，我们证明了$ k_ {ω，ω_1} $ - ZFC+MA+MA+$ \ neg $ CH下的$ K_ {ω，ω_1} $的制造商的获胜策略，以及在ZFC+CH下为Breaker的获胜策略。我们证明了$ k_ {ω_1} $ - 游戏的结果类似。在这里，制造商在ZF+DC+AD下具有获胜的策略，而Breaker再次在ZFC+CH下具有胜利。
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 $K_{ω,ω_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{ω_1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.
