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Knaster and friends II: the C-sequence number. (English) Zbl 07355297

Summary: Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the \(C\)-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of \(\mathsf{ZFC}\) and independence results about the \(C\)-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general \(C\)-sequence spectrum and uncover some tight connections between the \(C\)-sequence spectrum and the strong coloring principle \(\mathrm{U}(\dots)\), introduced in Part I of this series.

MSC:

03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E55 Large cardinals
06E10 Chain conditions, complete algebras
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