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 hbox {Spec}(mif). Here mif abbreviates maximal independent family. We show that:whenever kappa _1<cdots <kappa _n are finitely many regular uncountable cardinals, it is consistent that {kappa _i}_{i=1}^nsubseteq hbox {Spec}(mif);whenever kappa has uncountable cofinality, it is consistent that hbox {Spec}(mif)={aleph _1,kappa =mathfrak {c}}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\kappa _i,\\kappa _{i+1})\\cap \\hbox {Spec}(mif)=\\emptyset \\end{aligned}$$\\end{document}for each i, 1le i<n.