We give topological characterizations of filters F on such that the Mathias forcing MF adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

A. We prove that after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this remains to be the case in any further extension which has the Sacks property. We conclude that there are no P-points in the Silver model. In particular, it is possible to construct a model without P-points by iterating Borel partial orders. This answers a question of Michael Hrušák. We also show that the same argument can be used for the side-by-side product of Silver forcing. This provides a model without P-points with the continuum arbitrary large, answering a question of Wolfgang Wohofsky.2010 Mathematics Subject Classification. Primary: 03E35.

2016-11-02T18:49:00

We define and study two classes of uncountable ⊆ * -chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. We also indicate possible ways of developing a structure theory for towers based on classification of their Tukey types.

No abstract

The other authors dedicate this paper to Alan, who doesn't look a year over 59Abstract. We show that the existence of a homeomorphism between ω * 0

Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees remain unknown. In this paper, we investigate the big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages and demonstrate that they have finite big Ramsey degrees if and only if there are only finitely many relations of every arity. This is the first time that the finiteness of big Ramsey degrees has been established for an infinite-language random structure.

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 ω-hitting and ω-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.