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Abstract

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 ^{ \lt \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.