Partial strong compactness and squares

TL;DR

This paper explores the relationship between properties of partially strongly compact cardinals, filter extensions, and the failure of square principles, providing new insights into their interconnected set-theoretic implications.

Contribution

It establishes an equivalence between properties of strongly compact cardinals and compactness in infinitary logic, and improves bounds on the consistency strength of -compactness.

Findings

If certain filters can be extended to ultrafilters, then square principles fail for a range of regular cardinals.

The equivalence links filter properties with logical compactness and combinatorial principles.

Results improve the known lower bounds for the consistency strength of -compactness.

Abstract

In this paper we analyze the connection between some properties of partially strongly compact cardinals: the completion of filters of certain size and instances of the compactness of $\mathcal{L}_{\kappa,\kappa}$. Using this equivalence we show that if any $\kappa$-complete filter on $\lambda$ can be extended to a $\kappa$-complete ultrafilter and $\lambda^{<\kappa} = \lambda$ then $\square(\mu)$ fails for all regular $\mu\in[\kappa,2^\lambda]$. As an application, we improve the lower bound for the consistency strength of $\kappa$-compactness, a case which was explicitly considered by Mitchell.