Mathias-Prikry and Laver-Prikry type forcing

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.