Chain conditions of products, and weakly compact cardinals

Abstract.  The history of productivity of the $\kappa$-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 $\kappa>\aleph_1$, the principle $\square(\kappa)$ is equivalent to the existence of a certain strong coloring $c:[\kappa]^2\rightarrow\kappa$ for which the family of fibers $\mathcal T(c)$ is a nonspecial $\kappa$-Aronszajn tree.

The theorem follows from an analysis of  a new characteristic function for walks on ordinals, and implies in particular that if  the $\kappa$-chain condition is productive for a given regular cardinal $\kappa>\aleph_1$, then $\kappa$ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if $\kappa$ is a weakly compact cardinal, then the $\kappa$-chain condition is productive.

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Citation information:

A. Rinot, Chain conditions of products, and weakly compact cardinals, Bull. Symbolic Logic, 20(3): 293-314, 2014.