Abstract
We prove that under the assumption of a supercompact cardinal κ which is a limit of supercompact cardinals, for any increasing Σ2 function φ the set {∂<κ:∂ is at least φ(∂) supercompact, is strongly compact, yet is not fully supercompact} is unbounded in κ. We then use ideas of Magidor to show that under the hypotheses of a supercompact cardinal which is a limit of supercompact cardinals it is consistent for the least strongly compact cardinal κ0 to be at least φ(κ0) supercompact yet not to be fully supercompact, where φ is again an increasing Σ2 function which also meets certain other technical restrictions.
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