Winning the pressing down game but not Banach Mazur

Jakob Kellner, Matti Pauna, Saharon Shelah

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 ${\omega }_1$, bu…