Team games, hypergraph spaces, and projective Boolean algebras

Abstract

We modify the game Fuchino, Koppelberg, and Shelah used to characterize the κ-Freese-Nation property for a given Boolean algebra A, replacing players I and II each with a team of n players with limited information. We show that A is tightly κ-filtered exactly when team II has a winning strategy for every finite team size.Case κ=ℵ0 characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of “very I-favorable.” We similarly characterize Cohen algebras in terms of a team version of I-favorability.If A is the clopen algebra of the space of n-uniform hypergraphs on κ+n that avoid copies of [n+1]n, then team II has a winning strategy for our modified FKS game for team size n−1 but not n. For n≥3, this algebra also answers a question of Geschke when combined with a locally <κ-sized characterization of tightly κ-filtered Boolean algebras that we prove. Case κ=ℵ0 includes a locally finite characterization of projective Boolean algebras.