### 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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