The next best thing to a P-point

TL;DR

This paper investigates a specific ultrafilter on ^2 created via forcing, revealing its properties as a weak P-point, its position in the Rudin-Keisler and Tukey hierarchies, and its partition relations, advancing understanding of ultrafilter structures.

Contribution

It introduces a new ultrafilter constructed by forcing, analyzing its properties and hierarchy positions, and compares it with related ultrafilters, providing new insights into ultrafilter theory.

Findings

The ultrafilter is a weak P-point but not a P-point.

It is not basically generated and not Tukey-above [_1]^{<}.

It has continuum many ultrafilters Tukey-below it.

Abstract

We study ultrafilters on $\omega^2$ produced by forcing with the quotient of $\scr P(\omega^2)$ by the Fubini square of the Fr\'echet filter on $\omega$. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal 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 $[\omega_1]^{<\omega}$, 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 non-isomorphic selective ultrafilters.