Families of sets related to Rosenthal’s lemma

Archive for Mathematical Logic 58 (1-2):53-69 (2019)
A family \ is called Rosenthal if for every Boolean algebra \, bounded sequence \ of measures on \, antichain \ in \, and \, there exists \ such that \<\varepsilon \) for every \. Well-known and important Rosenthal’s lemma states that \ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than \\), the covering of category. We also study ultrafilters on \ which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters.
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DOI 10.1007/s00153-018-0621-8
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References found in this work BETA

Iterated Perfect-Set Forcing.James E. Baumgartner & Richard Laver - 1979 - Annals of Mathematical Logic 17 (3):271-288.
Selective Ultrafilters and Homogeneity.Andreas Blass - 1988 - Annals of Pure and Applied Logic 38 (3):215-255.
Combinatorics on Ideals and Forcing.Serge Grigorieff - 1971 - Annals of Mathematical Logic 3 (4):363-394.
Combinatorics for the Dominating and Unsplitting Numbers.Jason Aubrey - 2004 - Journal of Symbolic Logic 69 (2):482-498.

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