Distributive Aronszajn trees

Abstract

Ben-David and Shelah proved that if $λ$ is a singular strong-limit cardinal and $2^λ=λ^+$, then $\square^*_λ$ entails the existence of a normal $λ$-distributive $λ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_λ$ by $\square(λ^+,{<}λ)$. As $\square(λ^+,{<}λ)$ 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 $κ$ regular uncountable, $\square(κ)$ entails the existence of a partition of $κ$ into $κ$ 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 $ω_2$ cannot be split into two fat sets.