Partial strong compactness and squares

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.