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314:. If κ is subcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.)
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Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on
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Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is analogous to that of
299:. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many
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Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the
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303:. Existence of a 2-supercompact cardinal
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