The spectrum of independence

by Fischer and Shelah. [FiSh:1137]

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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