On Determinacy for Cut and Choose Games of Uncountable Length

Abstract

We obtain results on cut and choose games for complete Boolean algebras. Zapletal proved that there is a Boolean algebra $\mathbb{B}$ such that $\mathcal{G}_\omega^\textsf{candc}(\mathbb{B})$, the version of the game which ends on the $\omega$'th round, is undetermined. We prove that, assuming the consistency of a proper class of supercompact cardinals, the limit version $\mathcal{G}^{\textsf{candc}}_{< \lambda}(\mathbb{B})$, in which there are $\lambda$-many rounds but no concluding round, is consistently determined for all complete Boolean algebras $\mathbb{B}$ and all successor cardinals $\lambda$. In particular, this answers a question of Zapletal \cite[Question 2]{Zapletal1995}. We also show that undetermined instances of the game $\mathcal{G}^\textsf{candc}_\lambda(\mathbb{B})$ follow from the approachability property, extending results of Dobrinen, and we prove that undetermined instances are compatible with $\textsf{MM}^{++}$.