The spectrum of independence

Abstract

We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec (mif). Here mif abbreviates maximal independent family. We show that:1.whenever κ1< ⋯ < κn are finitely many regular uncountable cardinals, it is consistent that {κi}i=1n⊆Spec(mif);2.whenever κ has uncountable cofinality, it is consistent that Spec (mif) = { ℵ1, κ= c}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅for each i, 1 ≤ i< n.