The Next Best Thing to a P-Point

Abstract

We study ultrafilters on 2 produced by forcing with the quotient of P( 2 ) by the Fubini square of the Fréchet filter on . We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ 1 ] < , and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters. §1. Introduction. Quotients of the form P( )/I where I is an analytic ideal on are referred to as analytic quotients. Analytic quotients have been well-studied in the literature (see [11], [17], and [8]). These studies have usually focused on the structure of gaps in such quotients or on lifting isomorphisms between P( )/I and P( )/J , topics that are closely related. P( )/I is a Boolean algebra, and hence is a notion of forcing. 1 If forcing with an analytic quotient P( )/I does not add any new subset of , then the generic filter it adds is in fact an ultrafilter on that is disjoint from I. Ultrafilters added by analytic quotients have not been as extensively investigated, except for the most familiar analytic quotient P( )/Fin, which adds a selective ultrafilter. Recall the following definitions.Definition 1.1. An ultrafilter U on is selective if, for every function f : → , there is a set A ∈ U on which f is either one-to-one or constant. It is a P-point if, for every f : → , there is A ∈ U on which f is finite-to-one or constant.Indeed, selective ultrafilters can be completely characterized in terms of genericity over P( )/Fin -a well-known theorem of Todorcevic ([3]) states that in the presence of large cardinals an ultrafilter on is selective if and only if it is L(R), P( )/Fin)-generic. Similar characterizations were recently shown for a large class of ultrafilters forming a precise hierarchy above selective ultrafilters (see [6], [7], and [12]).The generic ultrafilter added by P( )/Fin has a simple Rudin-Keisler type as well as a simple Tukey type. Let F be a filter on a set X and G a filter on a set Y . Recall that we say that F is Rudin-Keisler(RK) reducible to G or Rudin-Keisler (RK)

Partitions and conservativity

“…In this section, we review the results from [4] that will play a major role in the present paper, and we take the opportunity to also fix some notation and terminology.…”

“…Part of the present paper is an analysis of the implications between these combinatorial properties in general, i.e., independently from the connection with the generic ultrafilters of [4]. This analysis involves also a notion of conservativity that was introduced by Phillips [9], was studied further in [1,2], and is connected with the model-theoretic notion of stability.…”

“…This paper is, in part, a sequel to an earlier joint paper with Natasha Dobrinen and Dilip Raghavan [4] in which we studied the ultrafilters created by a certain forcing construction. These ultrafilters were shown to have numerous combinatorial properties, including some weak partition properties and a classification of arbitrary functions modulo the ultrafilter.…”

“…The other part of the present paper concerns infinitary partition properties of the generic ultrafilters in [4]. These partition properties lead to so-called "complete combinatorics" for the forcing notion used in [4]. A different approach to these partition properties was developed independently by Dobrinen [6].…”

Banach spaces from barriers in high dimensional Ellentuck spaces

“…Moreover, the projection of U 2 to the first copy of ω recovers a Ramsey ultrafilter. In [6], many aspects of the ultrafilter U 2 were investigated, but the exact structure of the Tukey, equivalently cofinal, types below U 2 remained open. The topological Ramsey space E 2 and more generally the spaces E k , k ≥ 2, were constructed to produce dense subsets of P(ω k )/Fin k which form topological Ramsey spaces, thus setting the stage for finding the exact structure of the cofinal types of all ultrafilters Tukey reducible to the ultrafilter generated by P(ω k )/Fin k in [9].…”