Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters

Abstract: We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $\omega$-hitting and $\omega$-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.