Square in Core Models

Abstract

We prove that in all Mitchell-Steel core models, $\square_\kappa$ holds for all $\kappa$. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of $\square_\kappa$ if $\kappa$ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, $\square_\kappa$ holds iff $\kappa$ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)