Aronszajn trees, square principles, and stationary reflection

TL;DR

This paper explores the relationships between Aronszajn trees, square principles, and stationary reflection, showing how certain combinatorial principles are compatible or incompatible with stationary reflection and constructing special trees under specific conditions.

Contribution

It proves the compatibility of a weakened square principle with stationary reflection and establishes the existence of special trees at successors of singular cardinals under square assumptions.

Findings

Weaker square principle is compatible with stationary reflection.

Stronger square principle is incompatible with stationary reflection.

Square at a singular cardinal implies existence of special trees with ascent paths.

Abstract

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(\kappa)$ introduced by Brodsky and Rinot for the purpose of constructing $\kappa$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $\kappa$ but the stronger is not. We then prove that, if $\mu$ is a singular cardinal, $\square_\mu$ implies the existence of a special $\mu^+$-tree with a $\mathrm{cf}(\mu)$-ascent path, thus answering a question of L\"ucke.