Games with filters i

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 ω on κ is equivalent to weak compactness. Winning the game of length 2κ is equivalent to κ being measurable. We show that for games of intermediate length γ, II winning implies the existence of precipitous ideals with γ-closed, γ-dense trees. The second part shows the first is not vacuous. For each γ between ω and κ⁺, it gives a model where II wins the games of length γ, but not γ⁺. The technique also gives models where for all ω1 < γ ≤ κ there are κ-complete, normal, κ⁺-distributive ideals having dense sets that are γ-closed, but not γ⁺-closed.