We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martinʼs number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.

We study ultrafilters produced by forcing, obtaining different combinatorics and related Rudin-Keisler ordering; in particular we answer a question of Baumgartner and Taylor regarding tensor products of ultrafilters. Adapting a method of Blass and Mathias, we show that in most cases the combinatorics satisfied by the ultrafilters recapture the forcing notion in the Lévy model.