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We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinato-rial 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 does add a dominating real.(More)
The other authors dedicate this paper to Alan, who doesn't look a year over 59 Abstract. We show that the existence of a homeomorphism between ω * 0 and ω * 1 entails the existence of a non-trivial autohomeomorphism of ω * 0. This answers Problem 441 in [7]. We also discuss the joint consistency of various consequences of ω * 0 and ω * 1 being homeomorphic.
We outline a portfolio of novel iterable properties of c.c.c. and proper forcing notions and study its most important instantiations, Y-c.c. and Y-properness. These properties have interesting consequences for partition-type forcings and anticliques in open graphs. Using Neeman's side condition method it is possible to obtain PFA variations and prove(More)
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