More on the pressing down game

TL;DR

This paper explores the pressing down game and its connection to the Banach Mazur game, demonstrating the consistent existence of a specific ideal on  where the nonempty player wins under certain conditions.

Contribution

It shows, consistently, the existence of a nowhere precipitous normal ideal on  where the nonempty player wins the pressing down game of length , even when empty starts.

Findings

Existence of a nowhere precipitous normal ideal on .

Nonempty player can win the pressing down game of length .

Winning occurs even if the empty player starts.

Abstract

We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: Consistently, there is a nowhere precipitous normal ideal $I$ on $\aleph_2$ such that player nonempty wins the pressing down game of length $\aleph_1$ on $I$ even if player empty starts.