On strong $P$-points

Abstract

This paper investigates the combinatorial property of ultrafilters where the Mathias forcing relativized to them does not add dominating reals. We prove that the characterization due to Hrušák and Minami is equivalent to the strong P -point property. We also consistently construct a P -point that has no rapid Rudin-Keisler predecessor but that is not a strong P -point. These results answer questions of Canjar and Laflamme.

Canjar Filters

“…Laflamme noted without a proof that Canjar ultrafilters were strong P + -filters and asked if these two notions were equivalent. This was answered positively by Blass, Hrušák and Verner in [3]. We will now extend their result to the general case.…”

“…We say that an ideal I is a Canjar ideal if its dual filter I * = {ω − X | X ∈ I } is a Canjar filter. Canjar filters have been investigated in [7] and [3], this paper is a continuation of that line of research.…”

“…It was asked by Hrušák and Minami if every Borel Canjar ideal must be F σ and one of the main results of this article is to answer this question positively. In order to achieve this, we will extend a characterization of Canjar ultrafilters by Blass, Hrušák and Verner in [3].…”

The ultrafilter and almost disjointness numbers

“…In [18] it was proved that a filter is Canjar if and only if is has the Menger property (as a subspace of ℘ (ω)). Canjar filters have been further studied in [5], [28], [27], [23] and [30].…”

“…The Miller forcing PT consists of all Miller trees ordered by inclusion. 5 Miller forcing (also called "super perfect forcing") was introduced by Miller in [49], this is one of the most useful and studied forcings for adding new reals (see [2], [49] or [29]). By split (p) we denote the collection of all splitting nodes and by split n (p) we denote the collection of n-splitting nodes (i.e.…”