Laver ultrafilters

TL;DR

This paper introduces Laver ultrafilters, characterizes their combinatorial properties, compares them with other ultrafilter classes, and explores their generic existence within set theory.

Contribution

It defines Laver ultrafilters, provides combinatorial characterizations, and analyzes their position among known ultrafilter classes, including their existence conditions.

Findings

Laver ultrafilters properly contain rapid P-points.

They are properly contained in hereditarily rapid and measure zero ultrafilters.

It is consistent that P-points do not exist while Laver ultrafilters do.

Abstract

We introduce $\textit{Laver ultrafilters}$, namely ultrafilters $\mathcal{U}$ for which the associated Laver forcing $\mathbb{L}_{\mathcal{U}}$ has the Laver property. We give simple combinatorial characterisations of these ultrafilters, which allow us to analyse their position among several well-studied combinatorial classes, including $P$-points, rapid ultrafilters, and ultrafilters arising in Baumgartner's $\mathcal{I}$-ultrafilter framework. In particular, we show that the class of Laver ultrafilters properly contains the class of rapid $P$-points and that it is properly contained both in the class of hereditarily rapid- and in the class of measure zero ultrafilters. Finally, we investigate the (generic) existence of Laver ultrafilters and establish bounds on their generic existence number. In particular, we show that it is consistent that $P$-points do not exist while Laver ultrafilters exist generically.