Families of sets related to Rosenthal’s lemma

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Abstract

A family \(\mathcal {F}\subseteq \left[ \omega \right] ^\omega \) is called Rosenthal if for every Boolean algebra \(\mathcal {A}\), bounded sequence \(\big \langle \mu _k:\ k\in \omega \big \rangle \) of measures on \(\mathcal {A}\), antichain \(\big \langle a_n:\ n\in \omega \big \rangle \) in \(\mathcal {A}\), and \(\varepsilon >0\), there exists \(A\in \mathcal {F}\) such that \(\sum _{n\in A, n\ne k}\mu _k(a_n)<\varepsilon \) for every \(k\in A\). Well-known and important Rosenthal’s lemma states that \(\left[ \omega \right] ^\omega \) is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \({\wp (\omega )}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than \({{\mathrm{cov}}}(\mathcal {M})\), the covering of category. We also study ultrafilters on \(\omega \) which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters (and consistently selective ultrafilters comprise a proper subclass).

Keywords

Rosenthal’s lemma Ultrafilters Selective ultrafilters P-points Q-points 

Mathematics Subject Classification

Primary 28A33 28A60 03E17 Secondary 03E35 03E75 05C55 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The results of the paper come partially from author’s Ph.D. thesis [25] written under the supervision of Piotr Koszmider, whom the author would like to thank for the guidance, inspiring discussions and helpful comments.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.Kurt Gödel Research Center for Mathematical LogicUniversität WienWienAustria

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