Parametrizing topological Ramsey spaces. Yuan Yuan Zheng

Transcription

1 Parametrizing topological Ramsey spaces by Yuan Yuan Zheng A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Mathematics University of Toronto c Copyright 2018 by Yuan Yuan Zheng

2 Abstract Parametrizing topological Ramsey spaces Yuan Yuan Zheng Doctor of Philosophy Department of Mathematics University of Toronto 2018 We prove a general theorem indicating that essentially all infinite-dimensional Ramsey-type theorems proven using topological Ramsey space theory can be parametrized by products of infinitely many perfect sets. This theorem has applications in several known spaces, showing that certain ultrafilters are preserved under both side-by-side and iterated Sacks forcing. In particular, the well-known result of selective ultrafilters on the natural numbers are preserved under Sacks forcing is extended to the corresponding ultrafilters on richer structures. We also characterize ultrafilters in topological Ramsey spaces in an abstract setting. The technique of combinatorial forcing is crucial in the proof of the general parametrization theorem, and ultra-ramsey theory plays an important role in the applications. ii

3 Acknowledgements First and foremost, I would like to express my gratitude to my advisor Professor Stevo Todorcevic for his supervision. His enlightening comments have been a valuable guidance throughout this exploration. I am also grateful to Professor Natasha Dobrinen for taking the time to share her insightful thoughts and making many helpful remarks. It was great to be part of the Set theory community in Toronto. My fellow students, Fulgencio Lopez, Francisco Guevara Parra, Ming Xiao, Damjan Kalajdzievski, Jamal Kawach, and Sergio Garcia have made the Secret Student Set theory Seminar relaxing and enjoyable. The weekly Set theory Seminar at the Fields Institute has been a great source of information on cutting-edge research in the field. I am thankful to the postdoctoral fellows in the department, Diana Ojeda-Aristizabal, Yinhe Peng, Haim Horowitz, and Osvaldo Guzman Gonzalez, for sharing their perceptive insights. I also benefited from several illuminating discussions with Iian Smythe, David Fernandez Breton, Timothy Trujillo, and Professor Jose Mijares. During the time of my postdoctoral application, the support of Stevo and Natasha, as well as that of Professor Carlos Di Prisco, Professor Mary Pugh, and Professor Joe Repka, together helped smooth away my anxiety. I would also like to thank Anne Dranovski for proofreading my work and telling numerous fun facts including that the plural form of moose is not mooses or meese. Five years is a large portion of my life so far. Lastly I would like to thank myself for not giving up. However, this could not have been the case without my family, friends, and the staff in the Department of Mathematics at the University of Toronto. Many thanks to my parents, my grandma, and my best roommate Nikou for everything. iii

4 Contents Acknowledgements Table of Contents iii iv 1 Introduction Parametrizing Topological Ramsey Spaces Applications of the Parametrized Theorem Layout Preliminaries Sets and Colourings Topological Ramsey Spaces Example: the Ellentuck space Example: the Milliken space Sacks Forcing Ultrafilters in Topological Ramsey Spaces Sometimes Ramsey ultrafilters are selective Sometimes Nash-Williams ultrafilters are Ramsey Parametrized Theorems in Topological Ramsey Spaces The Condition (L4) (L4) in the Ellentuck space N [ ] Open Subsets of R R N Digression: Abstract Galvin Lemma Souslin-measurable Subsets of R R N Necessity of (L4) Ellentuck Space N [ ] Ultrafilters in the Ellentuck Space Ultrafilter quantifiers Equivalent properties for ultrafilters on ω Preservation of Ultrafilters under Sacks Forcing iv

5 5 Milliken Space FIN [ ] Ultrafilters in the Milliken Space Equivalent properties for ordered-union ultrafilters Parametrized Milliken Theorem Local Parametrized Milliken Theorem Open subsets of FIN [ ] (2 ω ) ω Perfectly U-Ramsey sets Selectivity Preserved under Side-by-side Sacks Forcing Selectivity Preserved under Iterated Sacks Forcing Spaces R α (α < ω 1 ) Topological Ramsey Spaces R α Ultrafilters in R α Parametrized R α Theorem Local Parametrized R α Theorem Open subsets of R α R N Perfectly U-Ramsey sets Preservation under Sacks Forcing Necessity of Nash-Williams High-dimensional Ellentuck Spaces High-dimensional Ellentuck Spaces E k Ultrafilters in E k Parametrized High-dimensional Ellentuck Theorem The Ultrafilter B k Conclusion and Further Discussions 73 Bibliography 75 v

6 Chapter 1 Introduction On my first day as a PhD student, I met the graduate administrator, Jemima Merisca, and asked if I had a pigeonhole in the department. After repeating the sentence but still seeing question marks popping out of Jemima s head, I realized that the word pigeonhole needed a replacement. To my surprise, the pigeonhole principle is called the pigeonhole principle. The infinite pigeonhole principle states that if infinitely many items are put into finitely many containers then there must be a container with infinitely many items inside. Comparing it to Ramsey s theorem a minor lemma Frank P. Ramsey [39] proved along the way to his goal of solving a problem in first-order logic we see that the pigeonhole principle is indeed the 1-dimensional Ramsey s theorem (d = 1). Throughout this thesis, by a colouring of a set S we mean a function c : S κ. We may call c a κ-colouring to emphasize the number of colours. If κ is finite, we may say that c is a finite colouring. Theorem 1.1 (Ramsey, [39]). For every positive integer d and every finite colouring of the family N [d] of all d-element subsets of the natural numbers, there is an infinite subset M of N such that the set M [d] of all d-element subsets of M is monochromatic. Ramsey s theorem bears the idea that within some sufficiently large systems, however disordered, there must be some order. A great amount of later work in mathematics was fruitfully developed out of Ramsey s theorem, which turned out to be an important early result in Combinatorics. The Ramseytype theorems on different structures developed out of Ramsey s theorem on the natural numbers are of the following form: for certain colourings of a mathematical structure, there exist monochromatic substructures of a particular type. Ramsey theory studies the conditions under which Ramsey-type theorems hold. Interestingly, the statement of Ramsey s theorem fails for d being infinite. For a set S, let S [ ] denote the set of all infinite subsets of S. The following example gives a 2-colouring of N [ ] such that there does not exist any infinite subset M of nnb with M [ ] being monochromatic. Example 1 (Folklore). In order to define the desired colouring c : N [ ] 2, first let E be an equivalence relation on N [ ] defined as follows. For X, Y N [ ], let X E Y if and only if their symmetric difference X Y is finite. Assuming the Axiom of Choice, let f : N [ ] /E N [ ] be a function which picks a 1

7 Chapter 1. Introduction 2 representative from each equivalence class. So for every X N [ ], the image of its equivalence class f([x] E ) is E-equivalent to X, i.e. f([x] E ) [X] E. Now define c by colouring an element X N [ ] according to the parity of the symmetric difference between X and the representative in the equivalence class containing X: for X N [ ], c(x) = 1 if and only if X f([x] E ) is odd. Thus, there does not exist any infinite subset M of nnb with M [ ] being monochromatic under the colouring c. Therefore, for infinite-dimensional Ramsey-type theorems to hold, restrictions must be imposed on colourings. The first result of infinite-dimensional Ramsey s theory is due to Nash-Williams [35]. Galvin- Prikry [21] showed that the infinite-dimensional Ramsey s theorem holds if the colouring is Borel with respect to the metric topology of N [ ]. Silver [41] generalized it to analytic colourings. In his new proof of Silver s theorem, Ellentuck [18] improved it to analytic colourings with respect to the exponential (or Vietoris) topology, which we now know as the Ellentuck topology. Ellentuck was the first to use topological notions to describe what is today generally considered the optimal form of this result, thus starting the whole area of topological Ramsey theory. We investigate finite colourings whose pre-images have certain topological properties. Souslin-measurable is the restriction imposed on colourings in the main theorem of this thesis (Theorem 1.7 below) and many theorems throughout this thesis. The definition of Souslin-measurable colourings is in Section 2.1. Topological Ramsey spaces were introduced to facilitate the study of higher-dimensional Ramsey-type theorems, and were axiomatized by Carlson-Simpson [9] and Todorcevic [43]. In such a space, Ramseytype theorems hold precisely when the colouring partitions the object into sets with the property of Baire relative to the topology associated to the space. In other words, the Ramsey property coincides with the property of Baire with respect to the Ellentuck topology in topological Ramsey spaces. Therefore, the results of Galvin-Prikry, Silver and Ellentuck have their abstract versions. Let R be a topological Ramsey space. Theorem 1.2 (Abstract Galvin-Prikry Theorem, [43]). Every metrically Borel subset of R is Ramsey. Theorem 1.3 (Abstract Silver Theorem, [43]). Every metrically Souslin-measurable subset of R is Ramsey. Theorem 1.4 (Abstract Ellentuck Theorem, [43]). If (R,, r) is a topological Ramsey space then every Ellentuck property of Baire subset of R is Ramsey and every Ellentuck meagre subset is Ramsey null. The relevant terms in Theorem 1.4 are defined in Section Parametrizing Topological Ramsey Spaces If X is a topological Ramsey space then we have the following Ramsey-type theorem. Theorem 1.5. For certain colourings of X there is a large subset Y X such that Y is monochromatic. It turns out that the Ramsey-type theorem one obtains from topological Ramsey space theory as above can, in some cases, be strengthened to the following parametrized form.

8 Chapter 1. Introduction 3 Theorem 1.6. For certain sets P and a certain colouring of X P there are large subsets Y X and Q P such that Y P is monochromatic. The first parametrization of an infinite dimensional Ramsey-type theorem was discovered by Miller [33] and Todorcevic [33, p183] in the 1980s. Since then, many authors ([36, 31, 32]) have parametrized Ramsey-type theorems with perfect sets of real numbers, i.e. P = R and Q R is a Cantor set. In the case of the Ellentuck space, this work has culminated by a discovery ([13, 10]) of the maximal parametrization with P and Q being infinite products of finite sets of prescribed sizes. This parametrization is maximal in the sense that it implies all possible parametrized theorems. This is because any stronger parametrization would be concerned with the monochromaticity of infinite subsets of the natural numbers, contradicting the fact that Ramsey s theorem fails when the dimension d is infinite, as shown in Example 1. However, not much is known about maximal parametrization of other topological Ramsey spaces. On the basis of the work of Di Prisco and Todorcevic [13] and using the method of combinatorial forcing ([21, 35]), we prove a general theorem (Theorem 1.7 below) which implies that essentially all infinite-dimensional Ramsey-type theorems proven using topological Ramsey theory can be parametrized by products of infinitely many perfect sets, provided that a moderate condition (L4) holds in the topological Ramsey space R. (L4) is a Ramsey-type condition which concerns itself with the colouring c : AR a +1 [a, A] {open subsets of (2 ω ) ω } of finite approximations of elements in R with open subsets of the product of perfect trees. It requires that one can obtain a monochromatic substructure by shrinking the set being coloured as well as the perfect trees. For the case R = N [ ], (L4) follows from the Halpern-Läuchli Theorem 3.3 ([25]). Discussions about (L4) can be found in Chapter 3. Theorem 1.7 (Moderately-abstract Parametrized Ellentuck Theorem). Let R be a topological Ramsey space satisfying (L4). For every finite Souslin-measurable colouring of R R N there exists A R and a sequence (P i ) i<ω of perfect sets of real numbers such that [, A] i<ω P i is monochromatic. This theorem has found applications in the analysis of set-theoretic forcing. Moreover, we expect the idea here can be similarly applied to solve a major open problem in the area: Question. Is maximal parametrization possible for other topological Ramsey spaces, as it is the case for the Ellentuck space? 1.2 Applications of the Parametrized Theorem In the 1990s, Todorcevic conjectured that many topological Ramsey spaces have ultrafilters associated to them analogous to the way selective ultrafilters are related to the Ellentuck space. In some cases, such ultrafilters have been studied in an independent context. For example, the ultrafilters associated to the Ellentuck space were first considered by Gustave Choquet in the 1960s, motivated by a problem in functional analysis, and were analyzed in great detail by other researchers in the 1970s. Mijares [30] gave a general notion of selective ultrafilters in topological Ramsey Ramsey spaces, which we now know as weakly selective ultrafilters. Later on, Di Prisco, Mijares and Nieto [11] defined selective ultrafilters in topological Ramsey spaces, and used them to prove that selective ultrafilters are (R, )-generic over L(R), where is the almost reduction relation on the topological Ramsey space R.

9 Chapter 1. Introduction 4 Forcing is a technique extensively used in mathematical logic and recursion theory. It was invented by Paul Cohen in the 1960s to extend the set-theoretic universe and prove consistency and independence results. However, enlarging the universe may destroy the properties of objects in the ground model. Selective ultrafilters in the Ellentuck space have the well-known nice property of being indestructible after Sacks forcing, product Sacks forcing and iterated Sacks forcing ([4, 3]). Also, selective ultrafilters in the Ellentuck space capture the Ramsey property of sets in the sense that, for certain colourings, a monochromatic substructure can be found as an element in the ultrafilter. It is then natural to ask if there exists a notion for ultrafilters in every topological Ramsey space so that the ultrafilters capture the Ramsey property and are preserved under Sacks forcings. In this thesis, we give a positive answer to this question in the Milliken space FIN [ ], a structure richer than the Ellentuck space. It turns out that the notion coincides with stable ordered-union ultrafilters Blass [5] introduced when he studied Hindman s theorem. The notion is also a special case of Mijares weakly selective ultrafilters and Di Prisco, Mijares and Nieto s selective ultrafilters. Hence we have the following theorem where P is Sacks forcing, side-by-side (product) Sacks forcing, or iterated Sacks forcing. Theorem 1.8. Let U be a selective ultrafilter on FIN in the ground model, and V a name for the upward closure {Ẏ FIN : [ ˇX] Ǔ [ ˇX] Ẏ } of U. Then V P is a selective ultrafilter. The situation becomes more complicated in other topological Ramsey spaces. In [17], Dobrinen and Todorcevic constructed a hierarchy of topological Ramsey spaces R α (α < ω 1 ) in order to classify the Tukey and Rudin-Keisler structure below U α (α < ω 1 ), where U α are ultrafilters introduced by Laflamme [24] to obtain different combinatorics and relate Rudin-Keisler ordering. We prove the following theorem and show that Nash-Williams ultrafilters in R α are preserved under countable-support side-by-side Sacks forcing P κ adding κ Sacks reals simultaneously. Theorem 1.9. Let U be a Nash-Williams ultrafilter in R α in the ground model, and V a name for the upward closure {Y : ( [X] U)([X] Y )} of U. Then Pκ ( V is a Nash-Williams ultrafilter in R α ). In particular, U α are preserved. Dobrinen, Mijares and Trujillo [15] showed that Nash-Williams ultrafilters capture the Ramsey property in a large class of topological Ramsey spaces. Another example is the hierarchy of high dimensional Ellentuck spaces E k (k < ω) Dobrinen [14] constructed to determine the Tukey type of ultrafilters G k forced by the analytic quotients P(ω k )/FIN k, answering a question left open by Blass, Dobrinen, and Raghavan [8]. It is known that selective ultrafilters in the Ellentuck space are minimal in the Tukey order, and Dobrinen proved that the initial Tukey type below each G k is a chain of size k. This poses the question: are these G k preserved under Sacks forcings, as selective ultrafilters are? Again, the answer is positive and we characterize the properties of ultrafilters in an abstract setting. But the exact relation among these properties remains to be seen. The work in the Milliken space utilises the parametrized Milliken theorem proved by Todorcevic [43], which we strengthen to a local version for selective ultrafilters. The localization of the parametrized theorem uses U-trees introduced by Blass [6] and ultra-ramsey theory developed in the works of Sirota and Louveau. In the spaces R α (α < ω 1 ) and E k (k < ω), the localization process was similar, but there

10 Chapter 1. Introduction 5 was not a parametrized Ramsey-type theorem available. Todorcevic used a richer space, the Hales- Jewett space, to code the product of the Milliken space with R N and obtained the parametrized Milliken theorem. However, it seems unlikely that a single topological Ramsey space is powerful enough to code every product space. Theorem 1.7 provides a solution to this problem, paving the way towards a proof of Todorcevic s conjecture mentioned at the beginning of this section. 1.3 Layout This thesis is divided into eight chapters, including this Introduction. Chapter 2 contains the preliminaries, including the axioms for topological Ramsey spaces, and the definition of Sacks forcing. We also discuss the properties of ultrafilters in topological Ramsey spaces. In Chapter 3 we present the proof of Theorem 1.7. Then in each of the remaining four chapters, we discuss a particular topological Ramsey space. Chapter 4 is a collection of well-known results about the Ellentuck space N [ ]. In particular, we include various definitions of selective ultrafilters in the space, and show that they are all equivalent in Proposition 4.5. Chapter 5 presents the Local Parametrized Milliken Theorem, from which we deduce the preservation of selective ultrafilters under (side-by-side and iterated) Sacks forcing. In Chapter 6, similar results regarding Nash-Williams ultrafilters are obtained in the spaces R α (α < ω 1 ). Chapter 7 concerns the high-dimensional Ellentuck spaces E k. The last chapter is the conclusion and further discussions on possible future directions of study. Most of the materials in Chapters 3 and 7 are included in [49]. The contents in Chapter 5 and Chapter 6 have mostly been published in [47] and [48] respectively.

11 Chapter 2 Preliminaries 2.1 Sets and Colourings This section includes some notational conventions about sets and colourings, as well as definition of basic topological properties of sets. For a set S, let S [< ] be the set of all finite subsets of S, and let S < be the set of all finite sequences of elements in S. Similarly, let S [ ] be the set of all infinite subsets of S, and let S be the set of all infinite sequences of elements in S. In these cases, we may use ω and interchangeably in the subscription. Let n N. The meaning of S [<n], S <n, S [n], S n is clear by the same logic. If X is a set (resp. sequence), let X be the size (resp. length) of X. Now suppose X is a sequence and k N. Let X k be the restriction of X to its first k elements if k X, and let X k = X if k > X. For two sequences a, b, we say a is an initial segment of b, and write a b, if there is k N such that a = b k. If a b and a b, we may write a b. We use ω and N interchangeably to denote the set of all natural numbers {0, 1, 2,... }. Note that, the symbol is also used as following: for a function f on a set S and a subset X of S, f X is the image of elements in X under f. As discussed in the Introduction (Chapter 1), a κ-colouring (or simply a colouring) of a set S is a function c : S κ. A colouring is finite if it is a κ-colouring where κ is finite. In this thesis, we consider finite colourings. To make precise the topological restrictions imposed on colourings, we include the definitions of some properties in topological spaces. Definition 2.1. Let S be a topological space and X be a subset of S. We say X is dense in S if X has non-empty intersection with every non-empty open subset of S; X is nowhere dense if it is not dense in any non-empty open subset of S; X is meagre if it is a countable union of nowhere dense sets; X has the property of Baire if it is the symmetric difference of an open set and a meagre set. Definition 2.2. For a set S, a field on S is a collection of subsets of S which includes the empty set and is closed under finite intersection. If a field is in addition closed under countable intersection, then it is a σ-field Let S be a topological space, the collection of Borel sets is the minimal σ-field containing all open subsets of S. 6

12 Chapter 2. Preliminaries 7 Definition 2.3. A Souslin-scheme is a family A = {A s : s ω < }. Applying the Souslin operation A to this Souslin-scheme gives the set A( A) = A f n. f ω n ω The collection of Souslin sets in a topological space S is the minimal field containing all open subsets of S which is closed under the Souslin-operation. In the Induction, we mentioned Borel colourings, analytic colourings, and Souslin-measurable colourings. In general: Definition 2.4. Let P be a topological property, S be a topological space, and k ω. We say a colouring c : S k is P -measurable, or simply P, if for every n k, c 1 (n) has the property P. 2.2 Topological Ramsey Spaces In this section, let us recall some definitions and theorems about topological Ramsey spaces from [43]. Consider a triple (R,, r) where R is a nonempty set, is a quasi-ordering on R and the restriction function r : R ω AR which maps an element X R to the sequence (r n (X) = r(x, n)) of finite approximations of X. Notation. Unless otherwise specified, we use letters A, B, C and X, Y, Z for elements in R; a, b, c for elements in AR; m, n for natural numbers. The Ellentuck topology of R is the topology generated by basic open sets of the form [a, B] = {A R : (A B) ( n < ω)(r n (A) = a)} for a AR and B R. A set of the form [a, B] is called a basic set in R. Let the first-difference metric ρ : R R R as follows. For i < n and r n (X) r n (Y ), ρ(x, Y ) = 1 2 n if r i(x) = r i (Y ). The metric topology is the topology generated by the first-difference metric ρ on R. Alternatively, it is the product topology when we identify an element X R with its sequence (r n (X)) n<ω of finite approximations, and consider R as a subset of AR N. It has basic open sets of the form [a] = {A R : ( n < ω)(r n (A) = a)} for a AR. Clearly from the basic open sets, the Ellentuck topology extends the metric topology. We will be referring to the metric topology unless otherwise specified. We say (R,, r) is closed if R is a closed subset of AR N when X R is identified with (r n (X)) n<ω AR N and AR N has the product topology. Notation. For a, b AR, we say a is an initial segment of b (or b is an end-extension of a), and write a b, if there exists B R and n m < ω such that a = r n (B) and b = r m (B); a b if a b as above and n < m. We also write a B if a = r n (B) for some n < ω. The length a of a is n if there exists

13 Chapter 2. Preliminaries 8 A R such that a = r n (A). If a < n, then r n [a, A] = {r n (B) : (a B) (B A)}. Let AR[a, A] = r n [a, A]. n<ω In particular, AR[, A] is the collection of all finite approximations of elements in R that are below A in the ordering. The notion depth B (a) formalizes the measure of how long an initial segment of B is required in order to provide the material to construct a. For the quasi-ordering fin on AR in (A2) below, we define depth B (a) = min{k : a fin r k (B)}, where we set min =. Notation. Let R be a topological Ramsey space. For n < ω and B R, [n, B] = [r n (B), B]. For each n < ω, AR n = {r n (A) : A R}. We are now ready for the axioms of a topological Ramsey space. Definition 2.5 ([43]). We say (R,, r) is a topological Ramsey space if it is closed and satisfies axioms (A1) to (A4). (A1) (1) r 0 (A) = for all A R. (2) A B implies r n (A) r n (B) for some n. (3) r n (A) = r m (B) implies n = m and r k (A) = r k (B) for all k < n. (A2) There is a quasi-ordering fin on AR such that (1) {a AR : a fin b} is finite for all b AR, (2) A B if and only if ( n)( m)r n (A) fin r m (B), (3) a, b AR [a b b fin c ( d c)(a fin d)]. (A3) (1) If depth B (a) < then [a, A] for all A [depth B (a), B]. (2) A B and [a, A] imply that there is A [depth B (a), B] such that [a, A ] [a, A]. (A4) If depth B (a) < and if O AR a +1, then there is A [depth B (a), B] such that r a +1 [a, A] O or r a +1 [a, A] O =. Example 2 (Ellentuck space, [43]). Let R be N [ ] := {M N : M is infintie}, which is the set of all infinite subsets of N; let the quasi-ordering be the usual subset relation on N [ ] ; for A N [ ], let r n (A) be the set of the smallest n elements of A. Then AR = N [< ] = {a N : a is finite}.

14 Chapter 2. Preliminaries 9 The quasi-ordering fin on AR also coincides with the subset relation. In [43], it is checked that the Ellentuck space (N [ ],, r) is a topological Ramsey space. We will investigate the Ellentuck space in Chapter 4. Example 3 (Milliken space, [43]). Let FIN be the set of all finite nonempty subsets of N, i.e. FIN = N [< ] \ { }. For x, y FIN, by x < y we mean max(x) < min(y). A block-sequence is a sequence X = (x n ) n<m of elements in FIN where M ω and x n < x n+1 for all n < M 1. Each x n is called a block. The sequence is an infinite block-sequence if M = ω, and it is a finite block-sequence if M < ω. Let R = FIN [ ], which is the set of all infinite block-sequences. For two (finite or infinite) blocksequences X, Y, we say X is a block-subsequence of Y if every block in X is a finite union of blocks in Y. The quasi-ordering on FIN [ ] is defined by X Y if X is a block-subsequence of Y. For X = (x i ) i<ω FIN [ ] and n < ω, let r n (X) = (x i ) i<n. Then AR = FIN [< ], which is the set of all finite block-sequences. The quasi-ordering fin on FIN [< ] also coincides with the relation of blocksubsequence. In [43], it is checked that the Milliken space (FIN [ ],, r) is a topological Ramsey space. We will investigate the Milliken space in Chapter 5. Definition 2.6 ([43]). A subset X of R is Ramsey if for every nonempty basic set [a, A] there is a B [a, A] such that [a, B] X or [a, B] X =. We say X is Ramsey null if it is Ramsey and the second alternative always holds. Recall the definition of the property of Baire and meagre sets from Section 2.1. Theorem 1.4 (Abstract Ellentuck Theorem, [43]). If (R,, r) is a topological Ramsey space then every Ellentuck property of Baire subset of R is Ramsey and every Ellentuck meagre subset is Ramsey null. We will use the technique of fusion in topological Ramsey spaces. Definition 2.7. [43, Definition 4.25]. A sequence ([n i, Y i ]) i<ω of basic sets in R is a fusion sequence if (n i ) i<ω is an unbounded increasing sequence of integers, and Y i+1 [n i, Y i ] for all i < ω. The fusion lim Y i of the sequence is the unique element Y R such that r ni (Y ) = r ni )Y i for all i < ω. Note that the fusion exists since R is a closed subset of AR N. By (A1) and (A2), Y is unique and Y [n i, Y i ] for all i < ω. Let us return to the examples of the Ellentuck space (N [ ],, r) and the Milliken space (FIN [ ],, r), and see what we could obtain from the Abstract Ellentuck Theorem Example: the Ellentuck space Applying the Abstract Ellentuck Theorem 1.4 in Example 2 to the Ellentuck space, we obtain the Ellentuck Theorem Theorem 2.8 (Ellentuck, [18]). Suppose X N [ ] has the property of Baire with respect to the Ellentuck topology. Then for every basic set [a, A] there is B [a, A] such that [a, B] is either included in or disjoint from X. Corollary 2.9. Let d N and N be an infinite subset of N. For every colouring c : N [d] 2 there exists an infinite subset M of N such that M [d] is monochromatic.

15 Chapter 2. Preliminaries 10 Proof. Define a subset X of N [ ] by X = {X N [ ] : c(r d (X)) = 0 or r d (X) N}. Whether or not an element X in R is in X depends only on its finite approximation r d (X). So X is metrically open. Therefore X is also open with respect to the Ellentuck topology, and has the property of Baire with respect to the Ellentuck topology. Applying the Ellentuck Theorem 2.8 to the basic set [, N], we obtain M [, N] such that [, M] X or [, M] X =. In other words, M is an infinite subset of N such that M [d] monochromatic of colour 0 or monochromatic of colour 1. The famous Ramsey s Theorem follows as a corollary. Theorem 1.1 (Ramsey, [39]). For every positive integer d and every finite colouring of the family N [d] of all d-element subsets of the natural numbers, there is an infinite subset M of N such that the set M [d] of all d-element subsets of M is monochromatic. Proof. Suppose there are l colours, and the colouring is given by c : N [d] {0,..., l 1}. We identify the colours 1,..., l 1 and define a 2-colouring c 1 on N [d] : for X N [d], 0 if c(x) = 0; c 1 (X) = 1 if c(x) {1,..., l 1}. By Corollary 2.9, there exists an infinite subset N of N such that N [d] is monochromatic under the colouring c 1, and has at most l 1 colours under the original colouring c. Thus, we have reduced the number of colours by shrinking the set N to its infinite subset N. By recursively applying the above argument at most l 1 times, it follows that there is an infinite subset M of N such that M [d] is monochromatic under c Example: the Milliken space Applying the Abstract Ellentuck Theorem 1.4 to the Milliken Space in Example 3, we obtain Milliken s Theorem. Theorem 2.10 (Milliken, [34]). Suppose X FIN [ ] has the property of Baire with respect to the Ellentuck topology. Then for every basic set [a, A] there is B [a, A] such that [a, B] is either included in or disjoint from X. Hindman s Theorem then follows from Milliken s Theorem in the same way as Ramsey s Theorem follows from the Ellentuck Theorem. Corollary 2.11 (Hindman s Theorem, [22, 2]). For every finite colouring of FIN there is B = (b n ) n<ω FIN [ ] such that the set r 1 [, B] = {y FIN : ( k < ω)( n 0 < < n k )(y = b n0 b nk )} is monochromatic.

16 Chapter 2. Preliminaries 11 Proof. Similar to the proof of Ramsey s Theorem 1.1, we consider a 2-colouring c : FIN 2. Define X FIN [ ] by X X if and only if c (r 1 (X)) = 0. Note that X is metrically open, so it is open with respect to the Ellentuck topology, hence X has the property of Baire with respect to the Ellentuck topology. Taking [a, A] = [, {{n} : n N}] and applying Milliken s Theorem 2.10, we obtain B FIN [ ] such that [, B] X or [, B] X =. So r 1 [, B] is monochromatic of colour 0 or monochromatic of colour 1, respectively. Via the mapping f : FIN N where f(x) = i x 2i, the more familiar version of Hindman s Theorem easily follows. Corollary 2.12 (Hindman s Theorem, [22].). For every finite colouring of N there exists M N [ ] such that the set { } x i : (k < ω) ( i < k)(x i M) i<k of all finite sums of elements in M is monochromatic. In the rest of this thesis, we will always refer to the metric topology of a topological Ramsey space unless stated otherwise. We may refer to a particular topological Ramsey space by R when the quasiordering and the restriction function r are understood from the context. 2.3 Sacks Forcing Let 2 ω and (2 ω ) ω be equipped with the product topology. We use 2 ω interchangeably with R, and (2 ω ) ω interchangeably with R N. As described in Section 2.1, on the set of all finite 01-sequences 2 <ω, the symbols, and respectively denote length of the sequence, initial segment and restriction to an initial segment of certain length. Two finite 01-sequences are comparable if one is an initial segment of the other; otherwise they are incomparable. Definition 2.13 ([3]). We call a nonempty set p 2 <ω a tree if it is -downwards closed. A tree p is perfect if every s p has incomparable end-extensions t, u p. In particular, every perfect tree is infinite. For a perfect tree p, let [p] = {f 2 ω : ( n ω)(f n p)} be the set of all infinite branches of p. Then [p] 2 ω is a perfect set. Definition 2.14 ([40]). Sacks forcing P is the set of all perfect trees, ordered by p q if p q. Note that p q if and only if [p] [q]. Definition 2.15 ([3]). For p P and s p, let p s = {t p : (t s) (s t)}. The number of branchings below s in the tree p is called the branching level of s in p, which is {i < s : ( t p)(( t > i) (t i = s i) (t (i + 1) s (i + 1)))}. The nth branching level l(n, p) of the tree p is the set of all s p which have branching level n and are -minimal with this property. Note that l(n, p) p is a collection of nodes in p, rather than a set of natural numbers. If p, q P, q p, n ω and l(n, q) = l(n, p), then we write q n p.

17 Chapter 2. Preliminaries 12 Lemma 2.16 (Fusion 1, [3]). Suppose (p k ) k ω P and (m k ) k ω ω is unbounded and increasing such that p k+1 m k p k for all k ω. Then q = k ω p k P and q m k p k for all k ω. We call (p k ) k ω a fusion sequence and q the fusion of the sequence. Now we are ready to define countable-support side-by-side Sacks forcing. Definition 2.17 ([3]). Let κ be an infinite cardinal. Let P k be the set of all sequences p = (p i ) i<κ such that, for every i < κ, p i P and for all but countably many i < κ, p i = 2 <ω. We say p i is the ith tree of p. For p = (p i ) i<κ and q = (q i ) i<κ in P κ, p q if p i q i for all i < κ. For p P κ, let [p] = i<κ [pi ]. For ε [p] and i < κ, let ε i be the ith component in ε, so ε i [p i ]. The support of p is supp(p) = {i < κ : p i 2 <ω }. So each p P κ has countable support. Definition 2.18 ([3]). Let κ be an infinite cardinal. Let n ω, p P κ, and F be a finite subset of κ. The set l(f, n, p) is a finite set of F -tuples defined as follows. l(f, n, p) = l(n, p i ). i F For σ l(f, n, p) and i F, let σ i denote the i th component of σ, so σ i p i. We may alternatively think of σ as a function σ : F 2 <ω such that σ(i) p i 2 <ω. When κ = ω, we often have the case F = {0,..., n 1} for some n ω. In such cases, l(f, n, p) = l(n, n, p), and we may write l(n, p) for short when there is no confustion. To make it clearer, we note that when p P, l(n, p) is a set of nodes as defined as in Definition 2.15; when p P κ, l(n, p) is short for l({0,..., n 1}, n, p), which is a set of n-tuples of nodes as defined in Definition For n ω, p, q P κ, and F κ finite, we write q F,n p if q p and q i n p i for all i F. Again, we may write q n p for q {0,...,n 1},n p. For n, F, p, q as above, and σ l(f, n, p), we write q σ p if q p and σ i p i for every i F. We also define p σ as follows. For i < κ, (p σ) i p i σ i if i F ; = p i otherwise. Moreover, let ε [p] and σ l(f, n, p). We say σ is a pre-initial segment of ε and ε is a post-endextension of σ, and write σ ε, if σ i ε i for every i F. Lemma 2.19 (Fusion 2, [3]). Let κ be an infinite cardinal. Let (p k ) k<ω P κ. Suppose that (F k ) k<ω is an increasing sequence of finite subsets of κ with k<ω supp(p k) k<ω F k; (m k ) k<ω is an unbounded increasing sequence of natural numbers. Suppose also that p k+1 F k,m k p k for every k < ω. Define q = (q i ) i<κ where q i = k<ω pi k for each i < κ. Then q P κ and q F k,m k p k for all k < ω. We call (p k ) k<ω a fusion sequence, and q the fusion of the sequence. In later chapters, we will construct fusion sequences. It is crucial to notice that, by the definition of P κ, every element of P κ has countable support. So k<ω supp(p k) in Lemma 2.19 is countable, and we can always find a sequence (F k ) k<ω of finite subsets of κ with k<ω supp(p k) k<ω F k. In particular, when κ = ω, k<ω supp(p k) ω k<ω m k. So we may take F k = m k in the construction of fusion sequences.

18 Chapter 2. Preliminaries 13 Recall that 2 ω has the product topology. So it has basic open sets of the form [2 <ω s] = {f 2 ω : s f}, for s 2 <ω. Let κ be an infinite cardinal. The set (2 ω ) κ also has the product topology, with basic open sets of the form [(2 <ω ) κ σ] = {ε (2 ω ) κ : σ ε}, where there is n < ω and F κ [< ] such that σ l(f, n, (2 <ω ) κ ). If κ = ω, we may think of such σ as an element of (2 <ω ) <ω. For p P κ, [p] inherits the subspace topology from (2 ω ) κ, with basic open sets of the form [p σ] for σ l(f, n, p). Notice that p σ is an element of P κ, so [p σ] = i<κ [(p σ)i ] as defined in Definition We keep in mind that denotes end-extension in different cases: Between finite approximations and elements in AR R, we use to denote end-extension of an element. The symbol is also used to denote end-extensions of a node inside a tree such as 2 <ω. 2.4 Ultrafilters in Topological Ramsey Spaces Definition Let S be a set. A (non-principal) ultrafilter on the base set S is a collection U of subsets of S with the following properties. For subsets M, N, A, B of S, (1) S U but {x} / U for all x S, (2) M N and M U implies that N U, (3) M = A B and M U implies that A U or B U, (4) M U and N U implies that M N U. There is also a general definition of ultrafilters in topological Ramsey spaces. By referring to an ultrafilter U in R, we understand that U is an ultrafilter in the space R according to Definition 2.21 below, rather than merely an ultrafilter on the base set R in the sense of Definition Definition 2.21 ([11]). A subset V R is an ultrafilter if the following holds. (a) V is a filter on (R, ), i.e. (i) A, B R ((A V) (A B) (B V)); (ii) For every A, B V and a AR, (([a, A] ) ([a, B] ) ( C V) (C [a, A] [a, B])). (b) V is a maximal filter on (R, ): If V is a filter on (R, ) and V V then V = V. (c) For every A V and a AR[, A], (i) if B [depth A (a), A] V then [a, B], (ii) if B V, B A, and [a, B], then there exists A [depth A (a), A] V such that [a, A ] [a, B]. Definition 2.22 ([43]). A family F AR is Nash-Williams if a b for every distinct pair a, b F. It is Sperner if a fin b for every distinct pair a, b F. Every Sperner family is Nash-Williams. The family AR n is Nash-Williams for every n < ω.

19 Chapter 2. Preliminaries 14 Notation. For a family F and an element X R, let F X = {Y F : Y X}. Definition Let R be a topological Ramsey space and U an ultrafilter in R. We say [17]. U is Nash-Williams if for every Nash-Williams family G AR and every partition G = G 0 G 1 there exists X U and i 2 such that G i AR[, X] = ; [30]. U is Ramsey if for all A U, a AR[, A] and n ω, and for every f : AR a +n 2 there exists B [depth A (a), A] U such that f is constant on r a +n [a, B]; [30]. U is weakly selective if for every A U and every {A b } b AR1 U A with [b, A b ] for each b AR 1 [, A] there exists B U A such that [b, B] [b, A b ] for every b AR 1 [, B]; [11]. U is selective if for every A U and every {A a } a A U A with [a, A a ] for each a AR[, A] there exists B U A such that [a, B] [a, A a ] for every a AR[, B]. It is clear from the definition that every selective ultrafilter is weakly selective. We prove in Section that these properties coincide for ultrafilters on N in the Ellentuck space N [ ]. Trujillo [46] constructed an ultrafilter which is weakly selective but not Ramsey in the space R 1 built by Dobrinen and Todorcevic [16]. But the definition of ultrafilters used in [46] differs from Definition 2.21 in that it does not satisfy 2.21 (c). We discuss the relations among these properties for ultrafilters in specific spaces in later chapters. For now we are interested in some general results Sometimes Ramsey ultrafilters are selective In [30], Mijares stated an extra axiom (A8) which guarantees that every Ramsey ultrafilter is weakly selective. Definition 2.24 ([30]). For a topological Ramsey space (R,, r) the axiom (A8) states that for arbitrary A, B R, n < ω and b r n [, B], r n+1 [b, B] r n+1 [b, A] [b, B] [b, A]. Theorem 2.25 ([30]). If (A8) holds in R then every Ramsey ultrafilter in R is weakly selective. Among the topological Ramsey spaces we are concerned with, the Ellentuck space N [ ], the Milliken space FIN [ ] and the spaces R α (α < ω 1 ) satisfy (A8), while the High-dimensional Ellentuck spaces E k (k 2) do not satisfy it, as we will see in Chapter Sometimes Nash-Williams ultrafilters are Ramsey We define a property of topological Ramsey spaces which guarantees that every Nash-Williams ultrafilter is Ramsey. Since subsets of AR of the form r a +n [a, B] are Nash-Williams families, the definition of Ramsey ultrafilters is very similar to that of Nash-Williams ultrafilters. The main difference is that we consider a fixed initial segment a AR for Ramsey ultrafilters, but not for Nash-Williams ultrafilters. Definition A topological Ramsey space R is initializable if the following conditions hold:

20 Chapter 2. Preliminaries 15 (1) For every element X R and n < ω, r n (X) r n+1 (X) and X = n<ω r n(x). (2) For every ultrafilter U in R, and for every B U and a AR, there exists A U such that a A and A \ a B. When the condition (1) holds, every element A R is an increasing union of its finite approximations, A = n<ω r n(a). So, for a b A, it makes sense to write A \ a and b \ a. Again, the Ellentuck space N [ ], the Milliken space FIN [ ], and the spaces R α (α < ω 1 ) are initializable while it is not yet clear whether the High-dimensional Ellentuck spaces E k (k 2) are initializable. Theorem If a topological Ramsey space R is initializable, then every Nash-Williams ultrafilter on R is Ramsey. Proof. Suppose R is initializable. Let U be a Nash-Williams ultrafilter in R. Let A U, a AR[, A] and n ω be given. Let f : AR a +n 2 be given. We define a partition AR a +n = G 0 G 1 as follows. For b AR a +n, there exists X R such that b = r a +n (X ). We abuse notation and write r a (b) for r a (X ). By (A1), this is independent of the choice of X. Then we let b G 0 if and only if X R with r a +n (X) = a (b \ r a (b)) and f(r a +n (X)) = 0. Since U is Nash-Williams, there exists B U and i 2 such that G i B =. As R is initializable, there exists B U with a B and B \ a B. Then by Definition 2.21 (a)(ii) of ultrafilters in R, because A, B U and [a, A], [a, B ], there exists B U such that B [a, A] [a, B ]. If G 0 B =, then for all b r a +n [, B], X R ((r a +n (X) = a (b \ r a (b))) (f(r a +n (X)) = 1)). Consider Y [a, B ]. We check f(r a +n (Y )) = 1. Since a Y, Y B B, and B \ a B, there exists b r a +n (B) such that r a +n (Y ) = a (b \ r a (b)). So f(r a +n (Y )) = 1. Since Y [a, B ] was arbitrary, we have f r a +n [a, B ] = 1. If G 1 B =, then b G 0 for all b r a +n [, B], i.e. X R with (r a +n (X) = a (b \ r a (b)) and f(r a +n (X)) = 0. In particular, f(a (b \ r a (b))) = 0 for all b r a +n [, B]. So f r a +n [a, B ] = 0.

21 Chapter 3 Parametrized Theorems in Topological Ramsey Spaces Miller [33] and Todorcevic ([33], p183) parametrized the Galvin-Prikry Theorem with perfect sets of real numbers. This is the first parametrization of an infinite-dimensional Ramsey-type theorem. Answering a question of Miller, Pawlikowski [36] extended this result to a parametrization of the Ellentuck Theorem. Thereafter, Mijares [31] parametrized the Abstract Ellentuck Theorem 1.4, for abstract topological Ramsey spaces, with perfect subsets of R. In this chapter, we aim to prove the main theorem, Theorem 1.7. It is a parametrization of the space with an infinite sequence of perfect subsets of the real numbers. The theorem can be applied to obtain preservation results related to countable-support side-by-side Sacks forcing, as we will see in later chapters. Theorem 1.7 (Moderately-abstract Parametrized Ellentuck Theorem). Let R be a topological Ramsey space satisfying (L4). For every finite Souslin-measurable colouring of R R N there exists A R and a sequence (P i ) i<ω of perfect sets of real numbers such that [, A] i<ω P i is monochromatic. We introduce the condition (L4) in Section 3.1. Sections 3.2 and 3.3 are devoted to the proof of Theorem 1.7. The necessity of (L4) in Theorem 1.7 is shown in Section 3.4. Before the discussion on (L4), we include a few definitions essential to the proof of the main theorem. The proof involves the notion of barriers, which was introduced by Nash-Williams [35] in the development of the theory of well-quasi-orderings. The notion has since had many applications far beyond its original use. Recall the definition of Nash-Williams family and Sperner family from Definition Definition 2.22 ([43]). A family F AR is Nash-Williams if a b for every distinct pair a, b F. It is Sperner if a fin b for every distinct pair a, b F. Definition 3.1. Let a AR and A R. We say F is a barrier on [a, A] if F is Sperner and every X [a, A] has an initial segment in F. We say F is a barrier on A if it is a barrier on [, A]. Definition 3.2 (Rank of barriers, [43]). Let R be a topological Ramsey space, a AR and A R. Let F be a barrier on [a, A]. Consider T (F) = {s AR[a, A] : (a s) ( t F)(s t)} 16

22 Chapter 3. Parametrized Theorems in Topological Ramsey Spaces 17 as a tree ordered by end-extension without infinite branches. We define a strictly decreasing map ρ F as follows. ρ F : T (F) Ord s sup{ρ F (t) + 1 : (t T (F)) (s t)}. The rank of F on [a, A] is rk(f) = ρ F (a). 3.1 The Condition (L4) Towards proving the Parametrized Infinite-Dimensional Ramsey Theorem ([43, Theorem 9.26]) about the maximal parametrization of the Ellentuck space, Todorcevic proved a lemma ([43, Lemma 9.7]) connecting the Ellentuck space and infinite products of finite sets of prescribed sizes. Our condition (L4) is a variant of this lemma for products of perfect sets. With the assumption of (L4), we are able to prove a parametrized version (Proposition 3.11) of the Abstract Galvin Lemma (Theorem 3.15). (L4) Let A R and a AR[, A]. Let {O b : b AR a +1 [a, A]} be a family of open subsets of (2 ω ) ω. Then there exists B [depth A (a), A], q P ω, and a clopen subset G of [q] such that O b [q] = G for all b AR a +1 [a, B] (L4) in the Ellentuck space N [ ] To get some intuition about (L4), we take (L4) in the Ellentuck space as an example. In the Ellentuck space, the set AR a +1 [a, A] in (L4) becomes the set of all natural numbers in the infinite set A which are greater than the maximal element in a. Without loss of generality, in the Ellentuck space, we may assume that the initial segment a in (L4) is empty. So (L4) in the Ellentuck space is equivalent to the following statement: Let M N [ ] and {O l : l M} be a family of open subsets of (2 ω ) ω. Then there exists an N M [ ] and q P ω such that there is a clopen subset G of [q] with O l [q] = G for all l N. We prove that (L4) in the Ellentuck space follows from the infinite Halpern-Läuchli Theorem below. Notation. For p P and n ω, by p(n) we denote the set of nodes in p with length n, i.e. p(n) = {s p : s = n}. Theorem 3.3 (HL ω, [25]). If p = (p i ) i ω P ω and n<ω j 2, A N [ ] and q = (q i ) i<ω p such that n A i<ω qi (n) G j. Corollary 3.4. For p P ω, n ω, and k<ω such that σ l(n, p) j 2 (q σ) i (k) G j. i<ω pi (n) = G 0 G 1, then there exists i<ω pi (n) = G 0 G 1 there exists A N [ ] and q n p k A i<ω Proof. Apply Theorem 3.3 to shrink p to q and ω to an infinite subset A, then further shrink q and A by applying Theorem 3.3 recursively, each time consider a single σ l(n, p) and have k A i<ω (q σ)i (k) included in one of G 0 and G 1. Since the set l(n, p) is finite, Theorem 3.3 is applied a finite number of times. Hence the resulting q is in P ω and the resulting A is an infinite set.

23 Chapter 3. Parametrized Theorems in Topological Ramsey Spaces 18 Before showing (L4) in the Ellentuck space, let us prove some properties of products of perfect sets. Lemma 3.5. For p P ω and O [p] open, there exists q p such that [q] O or [q] O =. Proof. If there exists ε (2 ω ) ω such that ε O [p], then since O is open, there exists a pre-initial segment σ (2 <ω ) <ω of ε such that [p σ] O. So let q = p σ. Otherwise, there exists no such ε hence [p] O =. Definition 3.6. Let p P ω, k ω, and S (2 ω ) ω. Let Θ be a set of pre-initial segments of elements in [p]. We say S [p] depends only on Θ if, for each σ Θ, either all or none of the post-end-extensions of σ are in S, i.e. either [p σ] S or [p σ] S =. A finite number of applications of Lemma 3.5 gives the following. Proposition 3.7. Let p P ω and n ω. Then for every open set O [p] there exists q n p such that O [q] depends only on l(n, q). Corollary 3.8. Suppose p P ω, n ω, and {O l : l ω} is a family of open subsets of (2 ω ) ω. Suppose also that (n l ) l<ω is a strictly increasing sequence of natural numbers above n. Then there exists q n p such that for every l ω, O l [q] depends only on l(n l, q). Proof. Starting with q 0 = p, we build a fusion sequence (q l ) l<ω as follows, such that q l+1 n l q l and O l [q l+1 ] depends only on l(n l, q l+1 ) for each l < ω: Suppose q l has been constructed. O l [q l ] is an open subset of [q l ]. by Proposition 3.7, there exists q l+1 n l l(n l, q l+1 ). This finishes the construction of the fusion sequence. q l such that O l [q l+1 ] depends only on Let q be the fusion of the sequence (q l ) l<ω. Then q n l+1 q l+1 for all l. As n l < n l+1, l(n l, q) = l(n l, q l+1 ). Since O l [q] O l [q l+1 ] and O l [q l+1 ] depends only on l(n l, q l+1 ), O l [q] also depends only on l(n l, q). Lastly, since q n0 p and n < n 0, q n p as required. Now we prove that (L4) holds in the Ellentuck space. Proposition 3.9. Suppose M N [ ] and {O m : m M} is a family of open subsets of (2 ω ) ω. Then for every p P ω and n ω there exists q n p, an infinite subset N M and a clopen subset G [q] such that O m [q] = G for every m N. Proof. Let (k m) m M {k ω : k > n} be an unbounded increasing sequence. Applying Corollary 3.8 to shrink p, we may assume that O m [p] depends only on l(k m, p) for each m M. For each m, we can find k m k m such that for every i k m, the nodes in p i (k m ) are end-extensions of those in l(k m, p i ). More precisely, i k m t p i (k m ) t l(k m, p i ) t t. Then O m [p] depends only on i k m p i (k m ) for all m M. Without loss of generality, (k m ) m M is also increasing. We define a partition c : k<ω i<ω pi (k) 2 as follows. Suppose k < ω and s = (s i ) i<ω is a sequence such that s i p i (k) for all i < ω. If k < k 0, let c( s) = 0. Otherwise there exists m M such that k [k m, k m+1 ). Let σ s = (s i k m ) i<km. Since O m [p] depends only on i<k m p i (k m ), [p σ s ] is either included or disjoint from O m. Let c( s) = 1 if and only if [p σ s ] O m.