Square compactness and Lindel\"of trees

TL;DR

This paper explores the properties of certain large cardinals and their relation to special types of trees, providing new characterizations and consistency results within set theory.

Contribution

It establishes that weakly square compact cardinals are strong limit cardinals and characterizes Lindel"of Aronszajn trees with no uncountable finitely branching subtrees.

Findings

Weakly square compact cardinals are strong limit cardinals

Lindel"of Aronszajn trees with no uncountable finitely branching subtrees are characterized topologically

The class of such trees lies strictly between Suslin and Aronszajn trees

Abstract

We prove that every weakly square compact cardinal is a strong limit cardinal. We also study Aronszajn trees with no uncountable finitely branching subtrees, characterizing them in terms of being Lindel\"of with respect to a particular topology. We prove that the class of such trees lies between the classes of Suslin and Aronszajn trees, and that the inclusions can consistently be strict.