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We give new, purely combinatorial characterizations of several kinds of large cardinals, such as strongly C(n)-compact and C(n)-extendible, in terms of reflecting measures. We then study the key property of tightness of elementary embeddings that witness strong C(n)-compactness, which prompts the introduction of the new large-cardinal notion of tightly C(n)-compact cardinal. Then we prove, assuming the Ultrapower Axiom, that a cardinal is tightly C(n)-compact if and only if it is either C(n−1)-extendible or a measurable limit of C(n−1)-extendible cardinals. In the last section we also give new characterizations of Σn-strong cardinals in terms of reflecting extenders.
Bagaria et al. (Mon,) studied this question.
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