Games with Filters

TL;DR

This paper explores Welch games, linking game outcomes to large cardinal properties and ideal structures, and constructs models demonstrating various winning strategies and ideal characteristics.

Contribution

It introduces Welch games and establishes their connection to large cardinals, providing models with specific winning strategies and ideal properties.

Findings

Winning strategies characterize large cardinals like weak compactness and measurability.

Existence of models where players win games of certain lengths but not longer.

Construction of models with ideals having specific closure and density properties.

Abstract

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^\kappa$ is equivalent to $\kappa$ being measurable. We show that for games of intermediate length $\gamma$, II winning implies the existence of precipitous ideals with $\gamma$-closed, $\gamma$-dense trees. The second part shows the first is not vacuous. For each $\gamma$ between $\omega$ and $\kappa^+$, it gives a model where II wins the games of length $\gamma$, but not $\gamma^+$. The technique also gives models where for all $\omega_1< \gamma\le\kappa$ there are $\kappa$-complete, normal, $\kappa^+$-distributive ideals having dense sets that are $\gamma$-closed, but not $\gamma^+$-closed.