Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces R α , α < ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space, and for each α < ω 1 , R α+1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for R α , and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R α , 2 ≤ α < ω 1 . These form a hierarchy of extensions of the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U α , for each 2 ≤ α < ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to U α is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U α form a descending chain of rapid p-points of order type α + 1.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4628 NATASHA DOBRINEN AND STEVO TODORCEVIC equivalence relations for uniform barriers on R 1 of rank n. In this paper, we prove similar results for all 2 ≤ α < ω 1 . In Theorem 4.22, we show that for all 2 ≤ α < ω 1 , for any uniform barrier B on R α of finite rank and any equivalence relation E on B, there is an X ∈ R α such that E restricted to the members of B coming from within X is exactly one of the canonical equivalence relations. For finite α, there are finitely many canonical equivalence relations on uniform barriers of finite rank; these are represented by a certain collection of finite trees. Moreover, the numbers of canonical equivalence relations for finite α are given by a recursive function. For infinite α, there are infinitely many canonical equivalence relations on uniform barriers of finite rank, represented by tree-like structures. These theorems generalize the Erdős-Rado Theorem [8] for uniform barriers of finite rank on the Ellentuck space, namely, those of the form [N] n .In the main theorem of this paper, Theorem 4.12, we prove new Ramsey-classification theorems for all barriers (and moreover all fronts) on the topological Ramsey spaces R α , 2 ≤ α < ω 1 . We prove that for any barrier B on R α and any equivalence relation on B, there is an inner Sperner map which canonizes the equivalence relation. This generalizes our analogous theorem for R 1 in [5], which in turn generalized the Pudlak-Rödl Theorem [13] for barriers on the Ellentuck space. These classification theorems were motivated by the following.Recently the second author (see Theorem 24 in [14]) has made a connection between the Ramsey-classification theory (also known as the canonical Ramsey theory) and the Tukey classificati...

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN. U U min,max z z t t t t t t t t t $

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl Theorem to this class of topological Ramsey spaces.To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor in [2] generating p-points which are k-arrow but not k + 1-arrow, and in a partial order of Blass in [3] producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra P(n). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space's associated ultrafilter have the structure exactly [ω] <ω . In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R 1 that in its complexity comes immediately after the classical Ellentuck space [5]. Associated with R 1 is an ultrafilter U 1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R 1 . This is analogous to the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U 1 : Every ultrafilter which is Tukey reducible to U 1 is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of U 1 , namely the Tukey type of a Ramsey ultrafilter. 1 2 NATASHA DOBRINEN AND STEVO TODORCEVICRecently the second author (see Theorem 24 in [15]) has made a connection between the Ramsey-classification theory (also known as the canonical Ramsey theory) and the Tukey classification theory of ultrafilters on ω. More precisely, he showed that selective ultrafilters realize minimal Tukey types in the class of all ultrafilters on ω by applying the Pudlak-Rödl Ramsey classification result to a given cofinal map from a selective ultrafilter into any other ultrafilter on ω, a map which, on the basis of our previous paper [4], he could assume to be continuous. Recall that the notion of a selective ultrafilter is closely tied to the Ellentuck space on the family of all infinite subsets of ω, or rather the one-dimensional version of the pigeon-hole principle on which the Ellentuck space is based, the principle stating that an arbitrary f : ω → ω is either constant or is one-to-one on an infinite subset of ω. Thus an ultrafilter U on ω is selective if for every map f : ω → ω there is an X ∈ U such that f is either constant or one-to-one on U. Since essentially any other topological Ramsey space has it own notion of a selective ultrafilter living on the set of its 1-approximations (see [12]), the argument for Theorem 24 in [15] is so general that it will give analogous Tukey-classification results for all ultrafilters of this sort provided, of course, that we have the analogues of the Pudlak-Rödl Ramsey-classification result for the corresponding topological Ramsey spaces.This paper is our first step towards a research in this direction. In particular, inspired by work of Laflamme [11], we build a topological Ramsey space R 1 , so that the ultrafilter associated with R 1 is isomorphic to the ultrafilter U 1 forced by Laflamme. In [11], Laflamme forced an ultrafilter, U 1 , which is weakly Ramsey but not Ramsey, and satisfies additional partition properties. Moreover, he showed that U 1 has complete combinatorics over the Solovay model. By work of Blass in [2], U 1 has only one non-trivial Rudin-Keisler equivalence class of ultrafilters strictly below it, ...

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)