The spectrum of independence

  • Vera FischerEmail author
  • Saharon Shelah
Open Access


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:
  1. 1.

    whenever \(\kappa _1<\cdots <\kappa _n\) are finitely many regular uncountable cardinals, it is consistent that \(\{\kappa _i\}_{i=1}^n\subseteq \hbox {Spec}(mif)\);

  2. 2.

    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
$$\begin{aligned} (\kappa _i,\kappa _{i+1})\cap \hbox {Spec}(mif)=\emptyset \end{aligned}$$
for each i, \(1\le i<n\).


Cardinal characteristics Independent families Spectrum Sacks indestructibility Ultrapowers 

Mathematics Subject Classification

03E17 03E35 



Open access funding provided by University of Vienna. Vera Fischer would like to thank the Austrian Science Fund (FWF) for the generous support trough Grant Y1012-N35. Saharon Shelah was partially supported by European Research Council Grant 338821 (this is paper 1137 on the author’s list).


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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Gödel Research CenterUniversity of ViennaViennaAustria
  2. 2.Einstein Institute of MathematicsThe Hebrew University of Jerusalem, Givat RamJerusalemIsrael

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