Square Compactness and the filter extension property

TL;DR

This paper investigates the consistency strength of a set-theoretic property called square compactness, establishing it is at least as strong as weak compactness but less than indescribability, improving previous bounds from 1973.

Contribution

It provides the first known improvement to the upper bound of strong compactness related to square compactness in set theory.

Findings

Square compactness has at least the consistency strength of weak compactness.

Square compactness is strictly less than indescribability.

This work refines the understanding of the hierarchy of large cardinal properties.

Abstract

We show that the consistency strength of $\kappa$ being $2^\kappa$-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juh{\' a}sz.