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:
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1.
whenever \(\kappa _1<\cdots <\kappa _n\) are finitely many regular uncountable cardinals, it is consistent that \(\{\kappa _i\}_{i=1}^n\subseteq \hbox {Spec}(mif)\);
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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
for each i, \(1\le i<n\).
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Acknowledgements
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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Fischer, V., Shelah, S. The spectrum of independence. Arch. Math. Logic 58, 877–884 (2019). https://doi.org/10.1007/s00153-019-00665-y
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DOI: https://doi.org/10.1007/s00153-019-00665-y