Distributive Aronszajn trees

Abstract

Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^{\lambda }={\lambda }^{+}$, then ${\square }_{\lambda }^{\ast }$ entails the existence of a normal $\lambda$-distributive ${\lambda }^{+}$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis ${\square }_{\lambda }^{\ast }$ by $\square ({\lambda }^{+},<\lambda )$. As $\square ({\lambda }^{+},<\lambda )$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $\kappa$ regular uncountable, $\square (\kappa )$ entails the existence of a partition of $\kappa$ into $\kappa$ many fat sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that ${\omega }_2$ cannot be split into two fat sets.

45 pages; improved and generalized some results, and streamlined the presentation

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