Distributive Aronszajn trees

Abstract

Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ 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 $\square(\lambda^+,{<}\lambda)$. 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.