Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

# Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

Natasha Dobrinen Department of Mathematics
University of Denver
2280 S Vine St
Denver, CO  80208 U.S.A.
http://web.cs.du.edu/~ndobrine
José G. Mijares Department of Mathematics
University of Denver
2280 S Vine St
Denver, CO  80208 U.S.A.
and  Timothy Trujillo Applied Mathematics and Statistics
Colorado School of Mines
Golden, CO  80401 U.S.A.
Dedicated to James Baumgartner, whose depth and insight continue to inspire
###### Abstract.

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 -arrow but not -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 many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra . 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 . 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.

###### 2010 Mathematics Subject Classification:
03E02, 03E05, 03E40,05C55, 05D10, 54H05
Dobrinen was partially supported by National Science Foundation Grant DMS-1301665 and Simons Foundation Collaboration Grant 245286
Mijares was partially supported Dobrinen’s Simons Foundation Collaboration Grant 245286

## 1. Introduction

The Tukey theory of ultrafilters has recently seen much progress, developing into a full-fledged area of research drawing on set theory, topology, and Ramsey theory. Interest in Tukey reducibility on ultrafilters stems both from the fact that it is a weakening of the well-known Rudin-Keisler reducibility as well as the fact that it is a useful tool for classifying partial orderings.

Given ultrafilters , we say that is Tukey reducible to (written ) if there is a function which sends filter bases of to filter bases of . We say that and are Tukey equivalent if both and . The collection of all ultrafilters Tukey equivalent to is called the Tukey type of .

The question of which structures embed into the Tukey types of ultrafilters on the natural numbers was addressed to some extent in [9]. In that paper, the following were shown to be consistent with ZFC: chains of length embed into the Tukey types of p-points; diamond configurations embed into the Tukey types of p-points; and there are many Tukey-incomparable selective ultrafilters. However, [9] left open the question of which structures appear as initial Tukey structures in the Tukey types of ultrafilters, where by an initial Tukey structure we mean a collection of Tukey types of nonprincipal ultrafilters which is closed under Tukey reducibility.

The first progress in this direction was made in [23], where applying a canonical Ramsey theorem of Pudlák and Rödl (see Theorem 13), Todorcevic showed that every nonprincipal ultrafilter Tukey reducible to a Ramsey ultrafilter is in fact Tukey equivalent to that Ramsey ultrafilter. Thus, the initial Tukey structure below a Ramsey ultrafilter is simply a singleton.

Further progress on initial Tukey structures was made by Dobrinen and Todorcevic in [10] and [11]. To each topological Ramsey space, there is a naturally associated ultrafilter obtained by forcing with the topological Ramsey space partially ordered modulo finite initial segments. The properties of the associated ultrafilters are inherited from the properties of the topological Ramsey space (see Section 4). In [10], a dense subset of a partial ordering of Laflamme from [16] which forces a weakly Ramsey ultrafilter was pared down to reveal the inner structure responsible for the desired properties to be that of a topological Ramsey space, . In fact, Laflamme’s partial ordering is exactly that of an earlier example of Baumgartner and Taylor in [2] (see Example 21). By proving and applying a new Ramsey classification theorem, generalizing the Pudlák-Rödl Theorem for canonical equivalence relations on barriers, it was shown in [10] that the ultrafilter associated with has exactly one Tukey type of nonprincipal ultrafilters strictly below it, namely that of the projected Ramsey ultrafilter, and similarly for Rudin-Keisler reduction. Thus, the initial Tukey and Rudin-Keisler structures of nonprincipal ultrafilters reducible to the ultrafilter associated with are both exactly a chain of length .

In [11], this work was extended to a new class of topological Ramsey spaces , which are obtained as particular dense sets of forcings of Laflamme in [16]. In [11], it was proved that the structure of the Tukey types of ultrafilters Tukey reducible to the ultrafilter associated with is exactly a decreasing chain of order-type . Likewise for the initial Rudin-Keisler structure. As before, this result was obtained by proving new Ramsey-classification theorems for canonical equivalence relations on barriers and applying them to deduce the Tukey and Rudin-Keisler structures below the ultrafilter associated with .

All of the results in [23], [10] and [11] produced initial Tukey and Rudin-Keisler structures which are linear orders, precisely, decreasing chains of some countable successor ordinal length. This led to the following questions, which motivated the present and forthcoming work.

###### Question 1.

What are the possible initial Tukey structures for ultrafilters on a countable base set?

###### Question 2.

What are the possible initial Rudin-Keisler structures for ultrafilters on a countable base set?

###### Question 3.

For a given ultrafilter , what is the structure of the Rudin-Keisler ordering of the isomorphism classes of ultrafilters Tukey reducible to ?

Question 3 was answered in [23], [10] and [11] by showing that each Tukey type below the associated ultrafilter consists of iterated Fubini products of p-points obtained from projections of the ultrafilter forced by the space.

Related to these questions are the following two motivating questions. Before [11], there were relatively few examples in the literature of topological Ramsey spaces. The constructions in that paper led to considering what other new topological Ramsey spaces can be formed. Our general construction method presented in Section 3 is a step toward answering the following larger question.

###### Question 4.

What general construction schemes are there for constructing new topological Ramsey spaces?

We point out some recent work in this vein constructing new types of topological Ramsey spaces. In [18], Mijares and Padilla construct new spaces of infinite polyhedra, and in [19], Mijares and Torrealba construct spaces whose members are countable metric spaces with rational valued metrics. These spaces answer questions in Ramsey theory regarding homogeneous structures and random objects. One of aims of the present work is to find a general framework for ultrafilters satisfying partition properties in terms of topological Ramsey spaces. See also [7] for a new construction scheme.

###### Question 5.

Is each ultrafilter on some countable base satisfying some partition relations actually an ultrafilter associated with some topological Ramsey space (or something close to a topological Ramsey space)? Is there some general framework of topological Ramsey spaces into which many or all examples of ultrafilters with partition properties fit?

Some recent work of Dobrinen in [7] constructs high-dimensional extensions of the Ellentuck space. These topological Ramsey spaces generate ultrafilters which are not p-points but which have strong partition properties; precisely these spaces yield the ultrafilters generic for the forcings Fin, . The structure of the spaces aids in finding their initial Tukey structures via new extensions of the Pudlák-Rödl Theorem for these spaces.

It turns out that whenever an ultrafilter is associated with some topological Ramsey space, the ultrafilter has complete combinatorics, meaning that in the presence of a supercompact cardinal, the ultrafilter is generic over . This was proved by Di Prisco, Mijares, and Nieto in [4], building on work of Todorcevic in [14] for the Ellentuck space. Thus, finding a general framework for ultrafilters with partition properties in terms of ultrafilters associated with topological Ramsey spaces has the benefit of providing a large class of forcings with complete combinatorics.

In this paper we provide a general scheme for constructing new topological Ramsey spaces. This construction scheme uses products of finite ordered relational structures from Fraïssé classes with the Ramsey property. The details are set out in Section 3. The goal of this construction scheme is several-fold. We aim to construct topological Ramsey spaces with associated ultrafilters which have initial Tukey structures which are not simply linear orders. This is achieved by allowing “blocks” of the members of the Ramsey space to consist of products of structures, rather than trees as was the case in [11]. In particular, for each , we construct a hypercube space which produces an ultrafilter with initial Tukey and Rudin-Keisler structures exactly that of the Boolean algebra . See Example 24 and Theorems 60 and 67.

We also seek to use topological Ramsey spaces to provide a unifying framework for p-points satisfying weak partition properties. This is the focus in Section 4. All of the p-points of Baumgartner and Taylor in [2] fit into our scheme, in particular, the -arrow, not -arrow p-points which they construct. In the other direction, for many collections of weak partition properties, we show there is a topological Ramsey space with associated ultrafilter simultaneously satisfying those properties.

The general Ramsey-classification Theorem 38 in Section 6 hinges on Theorem 31 in Section 5, which generalizes the Erdős-Rado Theorem (see Theorem 11) in two ways: by extending it from finite linear orders to Fraïssé classes of finite ordered relational structures with the Ramsey property and the Order-Prescribed Free Amalgamation Property (see Definition 29), and by extending it to finite products of members of such classes. Theorem 31 also extends the Product Ramsey Theorem of Sokič (see Theorem 14) from finite colorings to equivalence relations, but at the expense of restricting to a certain subclass of those Fraïssé classes for which his theorem holds. Theorem 31 is applied in Section 6 to prove Theorem 38, which generalizes the Ramsey-classification theorems in [10] for equivalence relations on fronts to the setting of the topological Ramsey spaces in this paper. Furthermore, we show that the Abstract Nash-Williams Theorem (as opposed to the Abstract Ellentuck Theorem) suffices for the proof.

Section 7 contains theorems general to all topological Ramsey spaces , not just those constructed from a generating sequence. In this section, general notions of a filter being selective or Ramsey for the space are put forth. The main result of this section, Theorem 56, shows that Tukey reductions for ultrafilters Ramsey for a topological Ramsey space can be assumed to be continuous with respect to the metric topology on the Ramsey space. In particular, it is shown that any cofinal map from an ultrafilter Ramsey for is continuous on some base for that ultrafilter, and even better, is basic (see Definition 48). This section also contains a general method for analyzing ultrafilters Tukey reducible to some ultrafilter Ramsey for in terms of fronts and canonical functions. (See Proposition 50 and neighboring text.)

Theorems 38 and 56 are applied in Section 8 to answer Questions 1 - 3. All initial Tukey and Rudin-Keisler structures associated with the ultrafilters generated by the class of topological Ramsey spaces constructed in this paper are found. Theorem 60, shows that whenever Fraïssé classes are used to generate a topological Ramsey space, then the initial Tukey structure below the associated ultrafilter is exactly the Boolean algebra . When infinitely many Fraïssé classes are used, then the initial Tukey structure of the p-points below the associated forced filter is exactly . In Theorem 66, we find the exact structure of the Rudin-Keisler types inside the Tukey types of ultrafilters reducible to the associated filter. Theorem 67 shows that if is a topological Ramsey space constructed from some Fraïssé classes , , and is a Ramsey filter on , then the Rudin-Keisler ordering of the p-points Tukey reducible to is isomorphic to the collection of all (equivalence classes of) finite products of members of the classes , partially ordered under embeddability.

The work in Sections 3 - 5 is due to Dobrinen. Section 6 comprises joint work of Dobrinen and Mijares. Sections 7 and 8 are joint work of Dobrinen and Trujillo, building on some of the work in Trujillo’s thesis. The main results in this paper for the special case of the space constitute work of Trujillo in his PhD thesis [27].

###### Acknowledgments.

The authors gratefully acknowledge input from the first anonymous referee pointing out an oversight in the first draft which led us to formulate the OPFAP. We also thank the second anonymous referee for pointing out some typos. Many thanks go to Miodrag Sockič for his thorough reading of previous drafts, catching typos and some errors which have been fixed.

## 2. Background on topological Ramsey spaces, notation, and classical canonization theorems

In [25], Todorcevic distills the key properties of the Ellentuck space into four axioms which guarantee that a space is a topological Ramsey space. For the sake of clarity, we reproduce his definitions here. The following can all be found at the beginning of Chapter 5 in [25].

The axioms A.1 - A.4 are defined for triples of objects with the following properties. is a nonempty set, is a quasi-ordering on , and is a mapping giving us the sequence of approximation mappings, where is the collection of all finite approximations to members of . For and ,

 (1) [a,B]={A∈R:A≤B and (∃n) rn(A)=a}.

For , let denote the length of the sequence . Thus, equals the integer for which . For , if and only if for some . if and only if for some . For each , .

1. for all .

2. implies for some .

3. implies and for all .

1. There is a quasi-ordering on such that

1. is finite for all ,

2. iff ,

3. .

We abuse notation and for and , we write to denote that there is some such that . denotes the least , if it exists, such that . If such an does not exist, then we write . If , then denotes .

1. If then for all .

2. and imply that there is such that .

If , then denotes the collection of all such that and .

1. If and if , then there is such that or .

The topology on is given by the basic open sets . This topology is called the Ellentuck topology on ; it extends the usual metrizable topology on when we consider as a subspace of the Tychonoff cube . Given the Ellentuck topology on , the notions of nowhere dense, and hence of meager are defined in the natural way. Thus, we may say that a subset of has the property of Baire iff for some Ellentuck open set and Ellentuck meager set .

###### Definition 6 ([25]).

A subset of is Ramsey if for every , there is a such that or . is Ramsey null if for every , there is a such that .

A triple is a topological Ramsey space if every property of Baire subset of is Ramsey and if every meager subset of is Ramsey null.

The following result can be found as Theorem 5.4 in [25].

###### Theorem 7 (Abstract Ellentuck Theorem).

If is closed (as a subspace of ) and satisfies axioms A.1, A.2, A.3, and A.4, then every property of Baire subset of is Ramsey, and every meager subset is Ramsey null; in other words, the triple forms a topological Ramsey space.

For a topological Ramsey space, certain types of subsets of the collection of approximations have the Ramsey property.

###### Definition 8 ([25]).

A family of finite approximations is

1. Nash-Williams if for all ;

2. Sperner if for all ;

3. Ramsey if for every partition and every , there are and such that .

The Abstract Nash-Williams Theorem (Theorem 5.17 in [25]), which follows from the Abstract Ellentuck Theorem, will suffice for the arguments in this paper.

###### Theorem 9 (Abstract Nash-Williams Theorem).

Suppose is a closed triple that satisfies A.1 - A.4. Then every Nash-Williams family of finite approximations is Ramsey.

###### Definition 10.

Suppose is a closed triple that satisfies A.1 - A.4. Let . A family is a front on if

1. For each , there is an such that ; and

2. is Nash-Williams.

is a barrier if (1) and () hold, where

1. is Sperner.

The quintessential example of a topological Ramsey space is the Ellentuck space, which is the triple . Members are considered as infinite increasing sequences of natural numbers, . For each , the -th approximation to is ; in particular, . The basic open sets of the Ellentuck topology are sets of the form and . Notice that the Ellentuck topology is finer than the metric topology on .

In the case of the Ellentuck space, the Abstract Ellentuck Theorem says the following: Whenever a subset has the property of Baire in the Ellentuck topology, then that set is Ramsey, meaning that every open set contains a basic open set either contained in or else disjoint from . This was proved by Ellentuck in [12].

The first theorem to extend Ramsey’s Theorem from finite-valued functions to countably infinite-valued functions was a theorem of Erdős and Rado. They found that in fact, given any equivalence relation on , there is an infinite subset on which the equivalence relation is canonical - one of exactly many equivalence relations. We shall state the finite version of their theorem, as it is all that is used in this paper (see Section 5).

Let . For each , the equivalence relation on is defined as follows: For ,

 b EI c⟺∀i∈I(bi=ci),

where and are the strictly increasing enumerations of and , respectively. An equivalence relation on is canonical if and only if there is some for which .

###### Theorem 11 (Finite Erdős-Rado Theorem, [13]).

Given , there is an such that for each equivalence relation on , there is a subset of cardinality such that is canonical; that is, there is a set such that .

Pudlák and Rödl later extended this theorem to equivalence relations on general barriers on the Ellentuck space. To state their theorem, we need the following definition.

###### Definition 12.

A map from a front on the Ellentuck space into is called irreducible if

1. is inner, meaning that for all ; and

2. is Nash-Williams, meaning that for all such that .

Given a front and an , we let denote . Given an equivalence relation on a barrier , we say that an irreducible map represents on if for all , we have .

The following theorem of Pudlák and Rödl is the basis for all subsequent canonization theorems for fronts on the general topological Ramsey spaces considered in the papers [10] and [11].

###### Theorem 13 (Pudlák/Rödl, [22]).

For any barrier on the Ellentuck space and any equivalence relation on , there is an and an irreducible map such that the equivalence relation restricted to is represented by .

Theorem 13 was generalized to a class of topological Ramsey spaces whose members are trees in [10] and [11]. In Section 6, we shall generalize this theorem to the broad class of topological Ramsey spaces defined in the next section.

## 3. A general method for constructing topological Ramsey spaces using Fraïssé theory

We review only the facts of Fraïssé theory for ordered relational structures which are necessary to this article. More general background on Fraïssé theory can be found in [15]. We shall call an ordered relational signature if it consists of the order relation symbol and a (countable) collection of relation symbols , where for each , we let denote the arity of . A structure for is of the form , where is the universe of , is a linear ordering of , and for each , . An embedding between structures for is an injection such that for any two , , and for all , . If is the identity map, then we say that is a substructure of . We say that is an isomorphism if is an onto embedding. We write to denote that can be embedded into ; and we write to denote that and are isomorphic.

A class of finite structures for an ordered relational signature is called hereditary if whenever and , then also . satisfies the joint embedding property if for any , there is a such that and . We say that satisfies the amalgamation property if for any embeddings and , with , there is a and there are embeddings and such that . satisfies the strong amalgamation property there are embeddings and such that and additionally, . A class of finite structures is called a Fraïssé class of ordered relational structures for an ordered relational signature if it is hereditary, satisfies the joint embedding and amalgamation properties, contains (up to isomorphism) only countably many structures, and contains structures of arbitrarily large finite cardinality.

Let be a hereditary class of finite structures for an ordered relational signature . For with , we use to denote the set of all substructures of which are isomorphic to . Given structures in , we write

 C→(B)Ak

to denote that for each coloring of into colors, there is a such that is homogeneous, i.e. monochromatic, meaning that every member of has the same color. We say that has the Ramsey property if and only if for any two structures in and any natural number , there is a with such that .

For finitely many Fraïssé classes , for some , we write to denote the set of all sequences such that for each , . For structures , , we write

 (Cj)j∈J→((Bj)j∈J)(Aj)j∈Jk

to denote that for each coloring of the members of into colors, there is such that all members of have the same color; that is, the set is homogeneous. We subscribe to the usual convention that when no appears in the expression, it is assumed that .

We point out that by Theorem A of Nešetřil and Rödl in [20], there is a large class of Fraïssé classes of finite ordered relational structures with the Ramsey property. In particular, the collection of all finite linear orderings, the collection of all finite ordered -clique free graphs, and the collection of all finite ordered complete graphs are examples of Fraïssé classes fulfilling our requirements. Moreover, finite products of members of such classes preserve the Ramsey property, as we now see. The following theorem for products of Ramsey classes of finite objects is due to Sokić and can be found in his PhD thesis.

###### Theorem 14 (Product Ramsey Theorem, Sokić [24]).

Let and be fixed natural numbers and let , , be a sequence of Ramsey classes of finite objects. Fix two sequences and such that for each , we have and . Then there is a sequence such that for each , and

 (Cj)j∈s→((B)j∈s)(Aj)j∈sk.

We now present our notion of a generating sequence. Such sequences will be used to generate new topological Ramsey spaces.

###### Definition 15 (Generating Sequence).

Let and , , be a collection of Fraïssé classes of finite ordered relational structures with the Ramsey property. For each , if then let , and if then let .

For each and , suppose is some fixed member of , and let denote the sequence . We say that is a generating sequence if and only if

1. For each , .

2. For each and all , is a substructure of .

3. For each and each structure , there is a such that .

4. (Pigeonhole) For each pair , there is an such that

 (An,j)j∈Jk→(Am,j)j∈Jk(Ak,j)j∈Jk.
###### Remark.

Note that (3) implies that for each and each , for all but finitely many .

We now define the new class of topological Ramsey spaces which are the focus of this article.

###### Definition 16 (The spaces R(⟨Ak:k<ω⟩)).

Let and , , be a collection of Fraïssé classes of finite ordered relational structures with the Ramsey property. Let be any generating sequence. Let . is the maximal member of .

We define to be a member of if and only if , where

1. is some strictly increasing sequence of natural numbers; and

2. For each , is some sequence , where for each , .

We use to denote , the -th block of . Let denote , the collection of all -th blocks of members of . The -th approximation of is . In particular, . Let , the collection of all -th approximations to members of . Let , the collection of all finite approximations to members of .

Define the partial order on as follows. For and , define if and only if for each there is an such that and for all , .

Define the partial order on as follows: For and , define if and only if there are and such that , , and for each , for some .

For and , denotes the minimal such that , if such a exists; otherwise . Note that for , is equal to . The length of , denoted by , is the minimal such that . For , we write if and only if there is a such that . In this case, we say that is an initial segment of . We use to denote that is a proper initial segment of ; that is and .

###### Remark.

The members of are infinite squences which are isomorphic to the maximal member , in the sense that for each -th block , each of the structures is isomorphic to . This idea, of forming a topological Ramsey space by taking the collection of all infinite sequences coming from within some fixed sequence and preserving the same form as this fixed sequence, is extracted from the Ellentuck space itself, and was first extended to more generality in [10].

The above method of construction yields a new class of topological Ramsey spaces. The proof below is jointly written with Trujillo.

###### Theorem 17.

Let and , , be a collection of Fraïssé classes of finite ordered relational structures with the Ramsey property. For each generating sequence , the space satisfies axioms A.1 - A.4 and is closed in , and hence, is a topological Ramsey space.

###### Proof.

Let denote . is identified with the subspace of the Tychonov power consisting of all sequences for which there is a such that for each , . forms a closed subspace of , since for each sequence with the properties that each and , then is a member of . It is routine to check that axioms A.1 and A.2 hold.

(1) If , then . If , then and for each , there is an such that . For each , let be an element of such that is a substructure of isomorphic to . Let . Then , so .

(2) Suppose that and . Let . Then since . Let . Then and .

Suppose that , , and . Let . By (4) in the definition of a generating sequence, there is a strictly increasing sequence such that , for each . For each , choose some in such that the collection

 {⟨nki,(Xi,j)j∈Jn⟩:(Xi,j)j∈Jn∈((Ci,j)j∈Jn(An,j)j∈Jn)}

is homogeneous for . Infinitely many of these will agree about being in or out of . Thus, for some subsequence , there are such that letting , we have that is either contained in or disjoint from . ∎

We fix the following notation, which is used throughout this paper.

###### Notation 1.

For and , we write to mean that there is some such that and for some . For and , let denote the collection of all such that .

For , , and . denotes the tail of which is above every block in . denotes the members of which are above .

## 4. Ultrafilters associated with topological Ramsey spaces constructed from generating sequences and their partition properties

In this section, we show that many examples of ultrafilters satisfying partition properties can be seen to arise as ultrafilters associated with some topological Ramsey spaces constructed from a generating sequence. In particular, the ultrafilters of Baumgartner and Taylor in Section 4 of [2] arising from norms fit into this framework. We begin by reviewing some important types of ultrafilters. All of the following definitions can found in [1]. Recall the standard notation , where for , we write to denote that .

###### Definition 18.

Let be a nonprincipal ultrafilter.

1. is selective if for every function , there is an such that either is constant or is one-to-one.

2. is Ramsey if for each -coloring , there is an such that takes on exactly one color. This is denoted by .

3. is a p-point if for every family there is an such that for each .

4. is a q-point if for each partition of into finite pieces , there is an such that for each .

5. is rapid if for each function , there exists an such that for each .

It is well-known that for ultrafilters on , being Ramsey is equivalent to being selective, and that an ultrafilter is Ramsey if and only if it is both a p-point and a q-point. Every q-point is rapid.

Let be any topological Ramsey space. Recall that a subset is a filter on if is closed upwards, meaning that whenever and , then also ; and for every pair , there is a such that .

###### Definition 19.

A filter on a topological Ramsey space is called Ramsey for if is a maximal filter and for each and each , there is a member such that either or else .

Note that a filter which is Ramsey for is a maximal filter on , meaning that for each , the filter generated by is all of .

###### Fact 20.

Let be any generating sequence with . Each filter which is Ramsey for generates an ultrafilter on the base set , namely the ultrafilter, denoted , generated by the collection .

###### Proof.

Let denote the collection of such that for some . Certainly is a filter on , since is a filter on . To see that is an ultrafilter, let be given. Since is Ramsey for , there is a such that either or else . In the first case, ; in the second case, . ∎

One of the motivations for generating sequences was to provide a construction scheme for ultrafilters which are p-points satisfying some partition relations. At this point, we show how some historic examples of such ultrafilters can be seen to arise as ultrafilters associated with some topological Ramsey space constructed from a generating sequence, thus providing a general framework for such ultrafilters.

###### Example 21 (A weakly Ramsey, non-Ramsey ultrafilter, [2], [16]).

In [10] a topological Ramsey space called was extracted from a forcing of Laflamme which forces a weakly Ramsey ultrafilter which is not Ramsey. That forcing of Laflamme is the same as the example of Baumgartner and Taylor in Theorems 4.8 and 4.9 in [2]. is exactly , where each , the linear order of cardinality . is dense in the forcing given by Baumgartner and Taylor. Thus, their ultrafilter can be seen to be generated by the topological Ramsey space .

The next set of examples of ultrafilters which are generated by our topological Ramsey spaces are the -arrow, not -arrow ultrafilters of Baumgartner and Taylor.

###### Definition 22 ([2]).

An ultrafilter is -arrow if and for every function , either there exists a set such that , or else there is a set such that . is an arrow ultrafilter if is -arrow for each .

Theorem 4.11 in [2] of Baumgartner and Taylor shows that for each , there are p-points which are -arrow but not -arrow. (By default, every ultrafilter is -arrow.) As the ultrafilters of Laflamme in [16] with partition relations had led to the formation of new topological Ramsey spaces and their analogues of the Pudlák-Rödl Theorem in [10] and [11], Todorcevic suggested that these arrow ultrafilters with asymmetric partition relations might lead to interesting new Ramsey-classification theorems. It turns out that the constructions of Baumgartner and Taylor can be thinned to see that there is a generating sequence with associated topological Ramsey space producing their ultrafilters. In fact, our idea of using Fraïssé classes of relational structures to construct topological Ramsey spaces was gleaned from their theorem.

###### Example 23 (Spaces An, generating n-arrow, not (n+1)-arrow p-points).

For a fixed , let and denote the Fraïssé class of all finite -clique-free ordered graphs. By Theorem A of Nešetřil and Rödl in [20], has the Ramsey property. Choose any generating sequence . One can check, by a proof similar to that given in Theorem 4.11 of [2], that any ultrafilter on which is Ramsey for is an -arrow p-point which is not -arrow.

Let denote any ultrafilter on which is Ramsey for . It will follow from Theorem 67 that the initial Rudin-Keisler structure of the p-points Tukey reducible to is exactly that of the collection of isomorphism classes of members of , partially ordered by embedability. Further, Theorem 60 will show that the initial Tukey structure below is exactly a chain of length 2.

###### Remark.

In fact, Theorem A in [20] of Nešetřil and Rödl provides a large collection of Fraïssé classes of finite ordered relational structures which omit subobjects which are irreducible. Generating sequences can be taken from any of these, resulting in new topological Ramsey spaces and associated ultrafilters. (See [20] for the relevant definitions.)

The next collection of topological Ramsey spaces we will call hypercube spaces, , . The idea for the space was gleaned from Theorem 9 of Blass in [3], where he shows that, assuming Martin’s Axiom, there is a p-point with two Rudin-Keisler incomparable p-points Rudin-Keisler reducible to it. The partial ordering he uses has members which are infinite unions of -squares. That example was enhanced in [9] to show that, assuming CH, there is a p-point with two Tukey-incomparable p-points Tukey reducible to it. A closer look at the partial ordering of Blass reveals inside essentially a product of two copies of the topological Ramsey space from [10]. Our space was constructed in order to construct or force a p-point which has initial Tukey structure exactly the Boolean algebra . The spaces were then the logical next step in constructing p-points with initial Tukey structure exactly .

We point out that the space is exactly the space in [10].

###### Remark.

The space was investigated in [27]. All the results in this paper pertaining to the space are due to Trujillo.

###### Example 24 (Hypercube Spaces Hn, 1≤n<ω).

Fix , and let . For each and , let be any linearly ordered set of size . Letting denote the sequence , we see that is a generating sequence, where each is the class of finite linearly ordered sets. Let denote . It will follow from Theorem 60 that the initial Tukey structure below is exactly that of the Boolean algebra .

Many other examples of topological Ramsey spaces are obtained in this manner, simply letting be a Fraïssé class of finite ordered relational structures with the Ramsey property.

We now look at the most basic example of a topological Ramsey space generated by infinitely many Fraïssé classes. When , no longer suffices as a base for an ultrafilter. In fact, any filter which is Ramsey for this kind of space codes a Fubini product of the ultrafilters associated with for each index . However, the notion of a filter Ramsey for such a space is still well-defined.

###### Example 25 (The infinite Hypercube Space Hω).

Let . For each and , let be any linearly ordered set of size . Letting denote the sequence , we see that is a generating sequence for the Fraïssé classes being the class of finite linearly ordered sets. Let denote . It will be shown in Theorem 60 that the structure of the Tukey types of p-points Tukey reducible to any filter which is Ramsey for is exactly . The space is the first example of a topological Ramsey space which has associated filter with infinitely many Tukey-incomparable Ramsey ultrafilters Tukey reducible to it.

We point out that, taking and each , , to be the Fraïssé class of finite ordered -clique-free graphs, the resulting topological Ramsey space codes the Fubini product seen in Theorem 3.12 in [2] of Baumgartner and Taylor which produces an ultrafilter which is -arrow for all .

We conclude this section by showing how the partition properties of ultrafilters Ramsey for some space constructed from a generating sequence can be read off from the Fraïssé classes. Recall the following notation for partition relations. For , any , and an ultrafilter ,

 (2) U→(U)mk,l

denotes that for any and any partition of into pieces, there is a subset in such that is contained in at most pieces of the partition. We shall say that the Ramsey degree for -tuples for is , denoted , if for each , but .

It is straightforward to calculate the Ramsey degrees of ultrafilters Ramsey for topological Ramsey spaces constructed from a generating sequence, given knowledge of the Fraïssé classes used in the construction. For a given Fraïssé class , for each , let denote the number of isomorphism classes in of structures with universe of size . Let denote the collection of all finite sequences such that .

###### Fact 26.

Let , be a Fraïssé class of finite ordered relational structures with the Ramsey property, and be an ultrafilter Ramsey for for some generating sequence for . Then for each ,

 (3) R(UK,m)=Σs∈S(m)Πi<|s|Iso(K,si).
###### Examples 27.

For an ultrafilter Ramsey for the space , we have , , , and in general, .

For an ultrafilter Ramsey for the space , we have , , and . In fact, for each , , since the only relation is the edge relation. The numbers can be calculated from the recursive formula in Fact 26, but as they grow quickly, we leave this to the interested reader.

When , the Ramsey degrees are again calculated from knowledge of the Fraïssé classes and .

###### Fact 28.

For a topological Ramsey space constructed from a generating sequence for Fraïssé classes , , letting be an ultrafilter Ramsey for , we have

 (4) R(UR,2