Distributive Aronszajn trees
Volume 245 / 2019
Abstract
Ben-David and Shelah proved that if $\lambda $ is a singular strong-limit cardinal and $2^\lambda =\lambda ^+$, then $\square ^*_\lambda $ 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 $ by $\square (\lambda ^+,{ \lt }\lambda )$.
As $\square (\lambda ^+,{ \lt }\lambda )$ 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 classical 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.
