UC Berkeley

We develop and refine the theory of plus-one premice, first introduced by Neeman and Steel in \cite{PIPM} and \cite{FSPIPM}. This culminates in a Condensation Lemma for iterable plus-one premice. We then apply Condensation to the construction of $\square_\kappa$ sequences in these premice; this is similar to Schimmerling and Zeman's $\square_\kappa$ construction in \cite{zeman square proof}, but the presence of long extenders complicates both the techniques and the results. Our main result is that for plus-one premice with finitely many long generators, $\square_{\kappa , 2}$ holds exactly when $\kappa$ is neither subcompact nor the successor of a $1$-subcompact cardinal.