Chain conditions of products, and weakly compact cardinals

The history of productivity of the k-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 k > the principle □() is equivalent to the existence of a certain strong coloring c : [∗]2 k for which the family of fibers T(c) is a nonspecial K-Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the /c-chain condition is productive for a given regular cardinal ∗ > N |. then k is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if k is a weakly compact cardinal, then the /c-chain condition is productive.