The spectrum of independence

by Fischer and Shelah. [FiSh:1137]
Archive for Math Logic, February 2019
We study the set of possible size of maximal independent family, to which we refer as spectrum of independence and denote Spec (mi f). We show that: (1) whenever kappa_1 < ... < kappa_n are finitely many regular uncountable cardinals, it is consistent that {kappa_i}^n_{i = 1} subseteq Spec (mif) ; (2) whenever kappa has uncountable cofinality, it is consistent that Spec(mif) = {aleph_1, kappa = {c}} . Assuming large cardinals, in addition to (1) above, we can provide that (kappa_i, kappa_{i +1} ) cap Spec(mif) = emptyset for each i, 1 <= i < n .

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