Laver ultrafilters
Silvan Horvath, Tan Özalp
TL;DR
We define Laver ultrafilters as ultrafilters $\mathcal{U}$ for which the Laver forcing $\mathbb{L}_{\mathcal{U}}$ has the Laver property, and provide several equivalent combinatorial characterisations. The work situates these ultrafilters within the landscape of $P$-points, rapid ultrafilters, and $\mathcal{I}$-ultrafilters via $\mathcal{I}_f$-ultrafilters and Yorioka ideals, showing Laver ultrafilters are measure zero (hence nowhere dense) and enjoy robust closure properties such as Rudin–Keisler downward-closure and stability under sums. It derives bounds on the generic existence number $\mathfrak{ge}(\text{Laver})$ in terms of classical cardinal characteristics and analyzes their behavior across standard forcing models, proving Cohen models yield generically existing Laver ultrafilters while several other canonical models do not. Finally, it demonstrates consistency results including models with no $P$-points but with generic Laver ultrafilters and poses open questions about the precise boundaries of Laverness in various frameworks.
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.