The independence of GCH and a combinatorial principle related to Banach-Mazur games

Abstract

It was proved recently that Telgársky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\mathsf{GCH}+\square$. The proof introduces a combinatorial principle that is shown to follow from $\mathsf{GCH}+\square$, namely: $▽$: Every separative poset $\mathbb{P}$ with the $\hat{I}º$-cc contains a dense sub-poset $\mathbb{D}$ such that $|\{q\in \mathbb{D}:p\text{ extends }q\}|<\hat{I}º$ for every $p\in \mathbb{P}$. We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $▽$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $▽$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $▽$ holds with $\mathbb{P}$ equal to the weight-${\aleph }_{\ddot{I}}‰$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.