Power-collapsing games

Abstract

The game 𝔖_ls (κ) is played on a complete Boolean algebra 𝔹, by two players, White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p∈𝔹. In the α-th move White chooses p_α ∈(0,p)_𝔹 and Black responds choosing i_α ∈{0,1}. White wins the play iff \bigwedge_{β ∈κ} ⋁_{α ≥ β} p_α ^i_α =0, where p_α ⁰=p_α and p_α ¹=p∖ p_α . The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π (𝔹) or if 𝔹 contains a κ⁺-closed dense subset. On the other hand, if White has a w.s., then κ ∈ [𝔥₂(𝔹), π(𝔹)). The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2 ^{< κ} =κ ∈ Reg and forcing by 𝔹 preserves the regularity of κ, then White has a w.s. iff the power 2^κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S⊆ Reg there is a c.B.a. 𝔹 such that White (respectively, Black) has a w.s. for each infinite cardinal κ∈ S (resp. κ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game 𝔖_ls(κ) is undetermined.