Subcompact cardinals, squares, and stationary reflection

TL;DR

This paper explores the relationship between subcompact cardinals, square principles, and stationary reflection, establishing incompatibilities, forcing results, and optimal conditions within large cardinal frameworks.

Contribution

It generalizes Jensen's incompatibility results, shows how to force square principles under certain conditions, and refines the understanding of weak squares in relation to subcompactness.

Findings

Subcompactness precludes certain square principles.

Square can be forced to hold where no obstruction exists.

Results preserve other strong large cardinals.

Abstract

We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by alpha^+-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.