The Next Best Thing to a P-Point

Journal of Symbolic Logic 80 (3):866-900 (2015)
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Abstract

We study ultrafilters onω2produced by forcing with the quotient of${\cal P}$(ω2) by the Fubini square of the Fréchet filter onω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

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References found in this work

Selective ultrafilters and homogeneity.Andreas Blass - 1988 - Annals of Pure and Applied Logic 38 (3):215-255.
Cofinal types of ultrafilters.Dilip Raghavan & Stevo Todorcevic - 2012 - Annals of Pure and Applied Logic 163 (3):185-199.

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