STRONG COMPACTNESS, SQUARE, GCH, AND WOODIN CARDINALS

We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal $\kappa _0$ (the least measurable cardinal) exhibiting properties which are impossible when $\kappa _0$ is supercompact. In particular, we construct models in which $\square _{\kappa ^+}$ holds for every inaccessible cardinal $\kappa $ except $\kappa _0$ , GCH fails at every inaccessible cardinal except $\kappa _0$ , and $\kappa _0$ is less than the least Woodin cardinal.