Distributive Aronszajn trees

TL;DR

This paper proves that under certain set-theoretic conditions, the existence of a special type of Aronszajn tree can be derived using a new approach involving walks on ordinals and club guessing, expanding understanding of square principles.

Contribution

It demonstrates that the existence of a normal λ-distributive λ^+-Aronszajn tree follows from a weaker square principle, using novel methods involving walks and club guessing.

Findings

Replaces ^*_ with (^+,{<}) in proving the existence of Aronszajn trees.

Introduces a new construction method using walks on ordinals and club guessing.

Shows that (^+,{<}) does not bound the order-type of clubs, unlike previous assumptions.

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.