Winning the pressing down game but not Banach Mazur

Abstract

Let $S$ be the set of those $\alpha \in {\omega }_2$ that have cofinality ${\omega }_1$. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length ${\omega }_1$, but not the Banach Mazur game of length $\omega +1$ (both games starting with $S$).

New version: some improvements in presentation, references, history; main theorem slightly strengthened; some typos removed