Abstract

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. 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 Welch games. Player II having a winning strategy in the Welch game of length [math] on [math] is equivalent to weak compactness. Winning the game of length [math] is equivalent to [math] being measurable. We show that for games of intermediate length [math], II winning implies the existence of precipitous ideals with [math]-closed, [math]-dense trees. The second part shows the first is not vacuous. For each [math] between [math] and [math], it gives a model where II wins the games of length [math], but not [math]. The technique also gives models where for all [math] there are [math]-complete, normal, [math]-distributive ideals having dense sets that are [math]-closed, but not [math]-closed.