Chain conditions of Products, and Weakly Compact Cardinals

Abstract

The history of productivity of the κ -chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal $\kappa > \aleph _1 {\rm{,}}$ the principle □( k ) is equivalent to the existence of a certain strong coloring $c\,:\,[k]^2 \, \to $ k for which the family of fibers ${\cal T}\left( c \right)$ is a nonspecial κ -Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ -chain condition is productive for a given regular cardinal $\kappa > \aleph _1 {\rm{,}}$ then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ -chain condition is productive.