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We give topological characterizations of filters F on ω such that the Mathias forcing M F adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of
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)
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 . We also discuss the joint consistency of various consequences of ω * 0 and ω * 1 being homeomorphic.