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We study the Mathias--Prikry and the Laver type forcings associated withfilters and coideals. We isolate a crucial combinatorial property of Mathiasreals, and prove that Mathias--Prikry forcings with summable ideals are allmutually bi-embeddable. We show that Mathias forcing associated with thecomplement of an analytic ideal always adds a dominating real. We alsocharacterize filters for which the associated Mathias--Prikry forcing does notadd eventually different reals, and show that they are countably generatedprovided they are Borel. We give a characterization of $\omega$-hitting and$\omega$-splitting families which retain their property in the extension by aLaver type forcing associated with a coideal.

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