Topics in Ramsey Theory on Sets of Real Numbers

Justin Tatch Moore

A thesis submitted in conformïty with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics Ontario Institute for Studies in Education of the University of Toronto

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Abstract

The purpose of this thesis is to explore three topics in Ramsey theory of sets on the real numbers. The fist topic of discussion is a program was initiated by M. Souslin which focuses on investigating the relationships of chain conditions in topology. 1will dernonstrate that under the assumption

NI < add.(&) = c there is a c.c.c. nonseparable compactum which does not map onto [O, 1lWl. Other aspects of Souslin's program are also investigated. The second topic could be termed "continuous Ramsey theory." In this portion of the thesis we will study colorings c : [XI2 -+ _t wwhh are continuous and irreducible (c realizes every color on every large square). 1 will show that such colorings exist which are connected with unboundedness in (N,C) and ( ) The last topic is an investigation of the effects of random forcing on Ramsey theoretic statements about wl. We wiU investigate some of the applications to combinatorics and general topology. Acknowledgment s

Before embarking on the thesis at hand, 1 would Iike to make a few acknowl- edgments. First and foremost, 1 would like to thank Stevo Todoreevit for a fantastic supervision. He has spent countless hours of his time over the 1st t hr ee years cfiallenging, guiding, encouraging, ent ert aining, and educat- ing me in all forms of the word. He has always made the tirne to have a cup of coffee and discuss mathematics or anything else worth talking about. Ilijas Farah, while he was in Toronto, 1 felt was somewhat of a mathematical "big brother" to me. He always shared an interest in whatever 1 was working on and gave insight and suggestions whenever he could. His work and pure fascination with the Open Coloring hiom were quite infectious and led me to my work on Chapter 4. The Set Theory Seminar has also provided me with something 1 will never forget: "entertainment" every 1Vednesda.y at ?:IO PM sharp. It is truly a living seminar and 1 am gratefd for both the active and the passive roles I have been allowed to play in it. Beyond this, particular mathematicians are difficult to single out (as there are so many who have shared their time and Muenced me). One, however, sticks out and, 1 feel, deserves special notice. Marion Scheepers has always shared an interest in my work and offered me encouragement in my pursuits. 1have truly enjoyed (and benefited £rom) the large percentage of time which he has spent discussing mathematics with me whenever he happened to be within a 5km radius of my coordinates. 1 also wish to acknowledge the following financial sources. During the academic years 1996-98 1was supported by the University of Toronto Open Fellowship, which 1understand is comprised partly if not entirely of generous contributions from the grants of the faculty of the Mathematics department (1 believe that Frank Tall, Stevo TodorCeviC, and Bill Weiss perhaps helped me out more than the others, particularly in the later years). This fund- ing continued to a lesser degree in the academic years 1998-00 when 1 was supported by the Ontario Graduate SchoIarship. 1 also wish to thank Mima Dzamonja, Klaas Pieter Hart, and Stevo TodorCeviC for helping to cover my expenses when 1was visiting with them abroad. 1also received travel money via conferences £rom the NSF on a number of occasions-

Dedicated to Stephanie Rihrn Moore and her thesis in progress Gaudeamus

Contents

Introduction x

1 Souslinean Spaces 1 1.1 Local and Global Chain Conditions ...... 2 1.2 Souslin's Program ...... 4 1.3 Compacta, Boalean Algebras. Partial Orders. and Partitions . 5 1.4 Forcing Axioms ...... 8

Global Chain Conditions 10 2.1 Gaps and Souslinean Spaces ...... 12 2.2 Being Separable ...... 16 2.3 Souslin's Condition ...... 17 2.4 Being Linearly Fibered ...... 19 2.5 Helly's Condition ...... 20 2.6 An Example From Nt < adde(&) = c ...... 22 2.7 AnExampleFkornb>N2 ...... 27

3 Local Chain Conditions 30 3.1 Some Consequences K2 ...... 32 3.2 K3and Additivity of Measure ...... 35

4 Continuous Irreducible Colorings 39 4.1 A Coloring Associated With Measure ...... 41 4.2 The Alternation Map ...... 48 4.3 Continuous Coduigs for Real Numbers ...... 52

5 Measure Algebras 56 5.1 Measure Algebras ...... 57 5.2 Martin's Axiom and Random Graphs...... 59 5.3 A Variation of Maharam's Theorem ...... 62

6 Combinatorics 65

6.1 Partition Calculus ...... 66 6.2 Uniformizing Ladder Systems ...... 73 6.3 Wage's Lemma ...... 77 6.4 Fragments of Martin's Axiom ...... 80

7 Topology 82 7.1 Cornetrizable Spaces ...... 85 7.2 Extremally Disconnected Spaces ...... 87 7.3 Compact S Spaces ...... 89

7.4 Small Compactifications of L Spaces ...... 92

A Glossary Introduction

In 1930, Ramsey published his celebrated theorem [48] today known as Ram- sey's theorem. It states that if the n element subsets of N are split into finitely many pieces, then for some infinite set A C N, every n element subset of A is contained in one of the pieces. While it was ignored and lay dormant for some the, it was to eventually idluence a wide cross section of mathematics . Over time, various gener alizations and variations of Ramsey 's t heorem have been studied. In this thesis, we will be particularly interested in those variations which have a close cormection to the reds- The first result; of this kind was due to Sierpihski. He proved the following result, showing that an obvious generalization of Ramsey's theorem to the continuum is simply false.

Theorem. [52] There is a partition [RI2= Ko U KI of the unordered pairs of real numbers into two pieces such that whenever H is a ssubset of R and [Hl2 W contained in either Ko or Kithen H is countable.

While this might initially seem to render any study of Ramsey's theorem at the level of the continuum fruitless, this turns out to be far £rom the case. It is fiequently the case in general topology that we are presented with a partition [XI2 = KoU Ki of the pairs of an uncountable set X into two pieces Ko and KI in such a way that it is desirable to be able co fmd a large set H C X, all of whose pairs are in one of the pieces. If this was all we knew, then the partition mentioned above would indicate that indeed we can't hope for much. Oken, however, a particular partition has some additional properties which arise hem the situation at hand- Suppose, for instance, that we know that X is a collection of intervals in a linear order and that a pair of intervals (1,J) is in Koif 1 and J are disjoint and in KI otherwise. Whether or not such partitions have uncountable homogeneous sets turns out to be equivalent to a famous problem of Souslin

[58] and independent of the usual awioms of set theory (unlike Sierpiiiski's example above). The first three chapters of this thesis ni1l provide some of the history of Souslin's problem and the program it generated and some of the current research being done in this area. In Chapter 2 1 will focus on constructing compact spaces which behave like Souslin lines but which require only mild set theoretic assumptions.

Theorem. (NI< adde(&)= c) There is a compact space satisfying Souslin's condition which is nonseparable and does not map ont0 [O, 1lN1.

IC, (n < w) is a system of axioms which arose though a carefd analysis of Martin's Axiom its relation to the identification of chain conditions. A program has been launched to study which consequences of MAN, can be derived from the various axioms Kn. 1will make the following additions to the known results-

Theorem. (K2) Every set of reals of size Ni has measure 0.

Theorem. (&) Every union of NI measure O sets has measure 0.

Sierpi~%kï'spartition is dehed directly kom some well ordering of the continuum. The partition itself turns out to be a very wild subset of [RI2 fkom the point of view of measure and category. It is therefore natural to

ask when there are partitions [XI2 = KoU Kion a set of reals X which have properties sirnilar to those of Sierp%kils but such that Ko and Kl are, Say, open subsets of [XI2. It turns out that this question has a very different flavor to it and that there existence of such colorings depend largely on the size of X. This will be the topic of our study in Chapter 4. 1will demonstrate the folIowing.

Theorem. If X C RiN is 2Jf thin for some monotonie unbounded f then

there is a continuous map c : [XI2 + N such that if Y C X and d'[YI2 # N then Y is f thin.

Theorem. There Zs a 0-ideal of measure O sets N., a set X C NN of size add(N.), and a continuous coloring c : [XI2 + N such that c"[Y]~= N wheneuer Y X has the same cardinalzty as X.

Theorem. There is a continuous map alt : pl2 + N such that if X is O-directed and unbounded in (P, <*) then alt"[XI2 = N. The results of this chapter make some progress towards answerïng a problem of Zapletal asking whether OCA implies add(N) > N1 and a problem of TodorEeviC asking whether OCA implies c is N1. Sometimes, when considering applications, it is unnecessary to demand a hornogeneous "square" for a partition. Hereditary separability (HS) and the hereditary Lindelof (HL) property are two topological properties which are equident in the class of metric spaces. Whether these two properties can be equident in the class of ail regu1a.r topological spaces is a well known open problem in generd topology. It turns out that this question is very closely related to the following question which clearly has a Ramsey theoretic nature.

Question. 1251 If [w1I2 = KgU KI is a partition of the pazrs of the first uncountable cardinal, must there eiçist uncountable sets A and B such that for some i = 0,l the pair {a,,û) is in Kiwhenever a is in A, fl is in B and acp?

Whde the relationship between HS and HL is an interesthg topic of study in its own right, it turns out to have numerous applications in general topology and analysis. Many problems which seem unrelated to the relationship between HS and HL boil down to the above "rectangle problem" and hence do share a deep connection to this Ramsey theoretic consideration. In some cases it is very important to know how identwg HS and HL in a certain class of topological spaces affects the statu of some other combinatorial statement. In Chapters 5-7 we will consider the behavior of HS and HL in the presence of forcing with a measure algebra. A group of problems in general topobgy require knowledge of how close the properties HS and HL are to one another in the presence of the inequality 2'0 < 2,'. It turns out that one such problem is the foUowing question of Katëtov.

Question. (351 If X is a compact topological space and X2 is hereditarily normal, must X be metrizable?

Not only does the study of measure theoretic forcing seem as though it will shed light on this area, but this approach is somewhat unique in that it requires an analysis which is often more mathematical than metamathematical. Among the theorems 1will prove are the following.

Theorem. (MAe) W(9)holds after forcing with any measure algebra.

Here W(0) is a combinatorid statement studied by M. Wage because of its impact on the relationship between HS and HL.

Theorem. (MA,,) After forcing with any rneasure algebra

for al1 a! < wl.

Theorem. (MdN,)After forcing with a separable measwe algebra, HS and HL are equivalent in the class of cornetrizable spaces.

Theorem. After forcing with a homogeneous nonseparable measure algebra there is a Ostaszewski space. In particular HS dues not imply HL in the class of compact spaces. This last result answers a question of Todortevik and contrasts his result that MAH, implies "HL implies HS in the class of compact spaces after forcing with any measure algebra." Some of the materid in this thesis either has been published or dlbe published in the coming y-ear. Chapter 2 is essentidy a reproduction of [46]. The latter section of Chapter 3 and ail but the final section of Chapter 4 are reproduced 60m [45] with permission fiom Elsevier Science. Chapter 1

Souslinean Spaces

A well known theorem of Cantor and Dedekind [Il](see [12] for a translation) states that (Et, 5)is the only linear order which is

1. dense with no fkst or last element,

2, order complete, and

3. separable

In 1920 Souslin [58] asked whether separability could be relaxed to the following condition:

Souslin's Condition: Every family of pairwise disjoint intervals is countable.

It was known to Souslin that this problern is reaQ a question of whether Souslin's condition and separability are equivdent in the class of linear orders. While this question turned out to be undecidable in ZFC, this question has led to a number of developments in set theory and topology, both at mathematical level and at a metamathematical Levei and continues to fuel research to this day.

Some of the first progress on this problem was made by Kurepa in 1935. hi [38] Kurepa demonstrated that the theory of trees is quite close to the

theory of linear orders (see &O 1431, [53]). Here a tree (T, 4)is a partially ordered set in which all predecessors of an element of T are well ordered by

4. Kurepa showed that every linear order has a T-base B of intervals such that B is a tree when ordered by 2. He established that Souslin's problem is equivalent to the question of whether every tree in which all chains and antichains are countable is itself count able- This formulation of Souslin's problem is perhaps the one which is most commonly quoted today. Notice that this statement of Souslin's problem is purely Ramsey theoretic in nature.

1.1 Local and Global Chain Conditions

Further progress was made on Souslin's problem by Knaster and Szpilrajn in the Scottish Book (see Problem 192 in [44]). Knaster introduced the following condition in the context of compact topological spaces.

Knaster's Condition: Every uncountable family of open sets contains an uncountabIe linked fdy.

Here a family is Zinked if it has the pairwise intersection property. Notice that Souslin's condition naturally generalizes to topological spaces as well and that Knaster's condition clearly implies it. Knaster demonstrated that Souslin's problem is equivalent to whether Souslin's condition implies Knaster's condition in the clxs of linear orders (see [44]). Knaster, a geometer, was motivated by Helly's theorem [32] which states that a collection of intervals is linked if and only if it is centered. Knaster and Szpilrajn asked whether there is a topological space which satisfies Souslin's condition but not Knaster's condition. Another consideration investigated by Knaster and Szpilrajn in the Scot- tish Book was whether Souslin's condition is preserved by products. They remarked that if Souslin's condition is preserved in the product of two spaces, then it is preserved by arbitraq products. They &O showed that Knaster's condition is preserved by arbitrary products. In [54] Shanin, motivated by this result, introduced his own chah condition, which 1 will denote Shanin's Condition, and showed that it is also preserved by arbitrary products.

Shanin's Condition: Every uncountable family contains an uncountable centered family.

Chain conditions such as those of Souslin, Knaster, and Shanin are sometimes referred to as local chah conditions. They assert that any large enough fdy contains a large subcollection with some additional intersection properties. Global chain conditions, such as separability, were studied by Horn and Tarski in [33] in the context of von Neumann's problem (see problem 163 in [44]). These chah conditions assert that the open sets of a topological space can be decomposed into countably many fandies, each of which has some specified intersection properties. In compact spaces separability asserts, for instance, that the open sets of a topological space can be decomposed into countably many families each with the fimite intersection property.

1.2 Souslin's Program

The problem of when (and whether) topological spaces identify chah conditions has becorne a prominent theme in set theory and topology. In the 1960's and 1970 's, Jech, Solovay, and Tennenbaum completely resolved Souslin's problem by showing that it could neither be proven nor refuted using the usual axioms of set theory (see [34],[57], and [63]). Solovay and Tennen- baum's proof in [57] that Souslin's problem can consistently have a positive answer led to the consideration of an important new axiom known today as Martin's Axiom. While Souslin's problem has been resolved, his program remains - the study of chain conditions and their relationships. The focus of the next two chapters will be to explore two modern themes in tfiis program-

The est general program cm be considered as an analysis of the in- teraction between Souslin's condition and the global chah conditions. For example, if a topological space distinguishes between Souslin's condition and separability, what else can be said about the space? Must it map onto [O, Ilw1? Typically the analysis of such a question is carried out under a strong Baire category assumption such as PFA or MAN,.This is because it is well known to be consistent that objects such as Souslin lines do exist. In Chapter 2 we will investigate some of the limitations of this program by demonstrating how to build nonseparable compacta which satisfy Souslin's condition but which are in some sense "small." The second program is to analyze the relationships of the local chain conditions- This program has its roots in Knaster's original consideration of his chah condition, as well as in resdts of Kunen, Rowbottom, and Solovay concerning the identification of Souslin's condition and Shanin's condition under MAN, (an important variation of Martin's kiom). Current motivation for this program, however, cornes out of the result of TodorEeviC and VeliCkoviC in [83]which states that MAN,is that same as equating Souslin's condition and Shanin's condition in the class of compact topological spaces. What remains is the question of whether Shanin's condition can be relaxed

to some other local chah condition in this characterization (see problem BY of [24]). Chapter 2 will focus on some of the consequences of MAN, which can be proven by identifying Souslin's condition with a local chain condition such as Knaster 's condition.

1.3 Compacta, BooleanAlgebras, Partialor- ders, and Partitions

The last two sections of this chapter are intended to provide a little back- ground for some of chapters to corne. It turns out that discussions of topics related to Souslin's program can often be carried out in a number of different but essentially equïvalent contexts. The context which is most appropriate generally tends t O depend on history, established convention, and occasion- dyconvenience.

Definition 1.3.1. A Boolean algebra is a ring (B,+,-)with 1 such that a2 = a for every element of B.

If (8,+, .) is a Boolean algebra then one can dehe the operations

aAb = a-b

aVb = a+a-b+b aC = 1 + a.

It is routine to veriS. that A, V, and (-)' behave just as f~,U, and com- plementation would on a field of sets. This was explained by the foIlowing representation theorem of Stone.

Theorem 1.3.2. [59] For every Boolean algebra B there is a O-dimensional compact space X such that B is isomorphic to the collection of clopen sets C of X equip with the operations union and intersection (representing V and A respectively) .

The compact space in the previous theorem is called the Stone space of B and denoted st(B). It consists of the collection of ail maximal centered families (called ultrafilters) in B. It is easy to see that the chain conditions we dehed earlier have natural translations in the class of Boolean algebra. Thus, in addition to compact spaces, Boolean algebras are another context in which one can discuss Souslin's program.

If 23 is a Boolean dgebra then by defming a 5 b iff a A c = O whenever b A c = O we can equip 23 with a natural partial order (which becomes containment in st(B)). Thus a Boolean algebra is a specific instance of a partial order (P,5). With the Boolean algebraic example in mind, one can define G C P to be linked (centered) if every pair (finite subset) of element of G ha a Iower bound in P. If G iç centered and the lower bound can be found in G then G is cded a iilter. Again, all of the chah conditions mentioned above transfer to the class of partial orders as well. Moreover, for every partial order, there is a Boolean algebra (and hence a compact space) which is equivalent to it as far as the study of chain conditions is concerned (see [37]). Another special instance of a partial order mises £rom Ramsey theory. If S is a set and K is a subset of [SInthen we can consider the collection Q of aU nnite subsets H of S such that [Hln C K ordered by >. One can then dehe a Boolean algebra hmQ. Notice that the collection of all H S such that [Hln C K then this is a compact space when given the subspace topology koom P (S) (see [8O]). The translation of Souslin's condition in the class of partitions is "whenever there is an uncountable family 3 of disjoint fite sets F such that [FIn2 K,then there is a pair E # F in F such that

[E U FI" C K." Generally the terms used to represent the chah conditions remain the same in these classes. There ate some exceptions. Separability in the class of cornpacta is cded 0-centered in the class of Boolean algebras and partial orders. Souslin's condition, even in the class of cornpacta, is a somewhat dated term. 1have chosen to use it for the discussion of Souslin's program for aesthetic and historical reasons- It is better known today as the countable chain condition or c.c.c.. We will use the latter terminology when we are discussing partial orders in the later chapters of this thesis.

1.4 Forcing Axioms

In the course of solving the Continuum Problem, Cohen developed a general method known as "forcing" which has become widely used in producing independence results ever since its introduction in the 1960's. Jech and Ten- nenbaum used it to show that Souslin's problem cm have a negative answer. Solovay and Tennenbaum later used it to establish the consistency of a positive answer to Souslin's problem. Martin noticed that Solovay and Temenbaum's techniques could be used to prove the consistency of a more general fact about Baire category:

Martin's Axiom: Any compactum which satisfies Souslin's condition can not be covered by fewer than continuum many nowhere dense sets.

Martin's Axiom is always true if the Continuum Hypothesis holds- Solovay and Tennenbaum demonstrated that it is consistent with the negation of the Continuum Hypothesis as well. An important variation on this axiom which is actually more relevant to Souslin's program (for 0 = Hl)is the following.

MAe: Any compactum which satisfies Souslin's condition can not be covered by 0 nowhere dense sets.

Martin's Axiom and MAe &O have formulations in terms of partial orders and Boolean algebras. The former is probably the most useful. MAO:If (P,5) is a C.C.C. partial order and Va ((a < B) is a sequence of dense subsets of P then there is a filter G P which intersects each

Here 2) C P is dense if for every p in P there is a d in V such that d 5 p. Martin's Axiom is what is sometimes known as a forcing axiom. If one replaces the countable chah condition with some other condition in the definition of MAs then the result is a different forcing axiom. Probably the best known of these forcing axioms is the Proper Forcing Momor PFA. As even a brief introduction to PFA would be beyond the scope of this work, we leave this topic to the interested reader. For us it will be enough to know that PFA is a strengthening of MAN,which has vazious additional consequences which shall be mentioned at the appropriate junctures. Chapter 2

Analysis of Global Chain Condit ions

One direction which can be taken &om the solution of Souslin's problem is to ask how strong of a positive answer to Souslin's problem one can expect to prove is consistent. Since linear cornpacta are rather simple examples as far as compact topological spaces go, it is reasonable to ask how complex cornpacta must be if they distinguish between Souslin's condition and one of the global chah conditions such as separabiliw. Perhaps the simplest examples of cornpacta which make such a distinction are the large Tychonoff cubes. By Knaster and Szpilrajn's result, [O, 11' always satisfies Knaster's condition and hence Souslin's condition- On the other hand, if I is larger thân R, then [O, l]' is never separable. Treybig and Ward have shown (see [84]) that if L is a linear compactum and L maps continuously onto X x Y where X and Y are infinite cornpacta CHAPTER 2. GLOBAL CHALN CONDITIONS 11 then X and Y are metrizable. Thus linear compacta are rather far fkom mapping onto [O, 1IWland hence it is natural to ask whether it is consistent that al compacta which distinguish between Souslin's condition and one of the global chah conditions must map onto [O, 1Iw1. The following topological property is useful in understanding why certain compacta don't map onto [O, IlW1.

Defulition 2.0.1. A compact space K is said to be linearly fibered if there is a continuous map 4 : K -t [O, 11 such that the inverse images of points are linearly compact a.

Fact 2.0.2. If K is a compact Zinearly fibered space then K does not rnap onto [O, 1Iw1.

Proof. Clearly being a lineady fibered compactum is inherited to aU closed subspaces. Consequently every closed subset E of a linearly fibered compact space X contains a Linearly orderable subspace which is a Gs set. Since every linear compactum contains a point of countable n-character, E must contain a point of countable ?r-character and therefore X cannot map onto [O, 11'' by a well known result of Shapirovskiï[55]. Cl

The purpose of this chapter is to provide a general method for constructing a compactum fkom a set theoretic object known as a gap. The construction wiU be carried out in such a way that the combinatorial properties of the gap translate into various topological properties of the resulting compactum. In particular there are restrictions on the gap which ensure that the CKAPTER 2- GLOBAL CHAIN CONDITIONS 12

resulting space will be nonseparable, satisfy Souslin's condition, be hearly fibered, and so on. Moreover, the gaps required to yield relevant compacta require ody mild set theoretic assumptions (e.g. compatible with MANi). The bulk of this chapter will be devoted to the method .which converts a gap into a topological space and a discussion of how the properties of the gap used in the construction are reflected in the resulting compactum. It will frequently be more convenient for us to work with Boolean algebras and then consider their Stone spaces. Consequently, most of the arguments and language which follows is algebraic in nature. At the end of this chapter we will consider two applications of this method.

One of them wiLl be to produce what is essentially a ZFC example of a compactum which is Lneady fibered but which still distinguishes between Souslin's condition and separability. The second application uses the assumption b > Nî to bdd another linearly fibered compactum, this time differentiating between Souslin's condition and being o-linked.

2.1 Gaps and Souslinean Spaces

It is kequently the case in general topology that set theoretic objects prove to be rather useful in the construction of topological spaces. One example of such an object is a gap in the structure ([yw,E*). Classically speaking, a gap is a pair of sequences {aE : E < K) and {b, : î] < A) of infinite subsets of N such that CHAPTER 2. GLOBAL CWCONDITIONS

3. there does not exist a c N with a6 Ç* c and c n b, Gnite for all < n and 7 < A.

Gaps always exist by a standard application of Zorn's Lemma, but the existence of a gap with some additional properties quiddy becomes something which cannot be decided within ZFC (for further reading on gaps, see e.g. [79] or [74])- It turns out that gaps can be quite useful in defining topological spaces which bear some resemblance to Souslin lines (see, e.g. [9] and [66]). In this chapter we will present a generalization of a construction from an eady version of [66]. Before proceeding it is necessary to define some notation as weU as provide a rough framework for our Boolean algebra. The following gives us a more general formulation of the notion of a gap.

Definition 2.1.1. An ideal Z on PI is a subset of P (N) which is closed under takuig fhite unions and subsets. In addition Z is said to be a P-ideal if (1,C*) is o-directed (every countable set has an upper bound in Z). An ideal on N is dense if every infinite set contains an infinite subset in the ideal.

Definition 2.1.2. If a and b are subsets of N and Z is an ideal on N then a b abbreviates a \ b E Z and a II b abbreviates a n b E 1. A pair (A,B) of subsets of P(N)is said to be orthogonal modulo an ideal Z on N (or orthogonal in P(N)/Z) if a IZb whenever a is in A and b is in B. CHAPTER 2. GLOBAL CHAüV CONDITIONS 14

Definition 2.1.3. A subset c of N is a said to splzt a set r C A x B modulo an ideal Z if a sxc and b Irc whenever (a,b) is in r. If there does not exist a c which splits a subset r of A x B and A is orthogonal to B then r is said to be a gap modulo Z (or a gap in P(N)/Z). If I' = A x B then 1 will simply Say (A,B) is a gap modulo Z.

The advantage of th% more general definition is that is allows us to examine gaps which ate deihable - something which will be of use to us later. The typical defhability restriction on ideals is that they are analytic, i-e. the continuous image of a Polish space. It is &O that case that it is necessazy to consider gaps in algebras other than P(N)/fin in order to get results in ZFC (see the remark in [66] following Theorem 8.4). For more information on dehable gaps the reader is referred to [64]and [71]. From this point on (A,B) WU be a gap modulo an F, P-ideal Z. The ideal Z will moreover be generated by the collection of all finite changes of some compact set X Ç Z C P(N) (it is easy to verify that in fact all F, P-ide& are of this form) . 1wiU also assume that all fhite sets are in Z and that K is closed under subsets. The following definitions are modeled after those of an earlier version of [66] which dealt with the special case Z = fin and K = (0). Let

and define (s, m) < (t,n) to be end extension - that is m < n and tn [O, rn-

11 = S. For n E N, set lC, = {K f~[O, n - 11 : K E K). Instead of considering an arbitrary member of A x B,it will be necessary to restrict our attention Note that for every a E A, b B there is an n such that

is in A 8 B and hence A 8 B is also a gap modulo Z. In addition to A, B, Z, and IC, the parameters of our Boolean algebra will also include a subset r of A 8 B which also forms a gap modulo 2. If (a,b) E I' and (t,n) E T then define

1- T(ab) = {(s, m) E T : a \ s n [O,m - 11 E iC, and b n s E K~).

2- T&) = {(s,m) E T : ((s,m) < (t,n)) or ((t,n) < (s,m))).

F'rom this define the Boolean algebras

2. A = (T(,,): (t,n)E T)/h.

It is now useful to make some observations. First note that (Tl<) is a fhitely branching tree. The followïng two facts are usefd in dealing with elements of B.

Fact 2.1.4. a is in IC if and only if an [O, n - 11 is in ?Cn for al1 n.

Proof. This follows Tom the compactness of K.

Fact 2.1.5. If F is a positive element of B then there is a jînite collection of generators whose meet is positive and contained in F. CHAPTER 2, GLOBAL CHAllV CONDITIONS 16

Proof. Obsenre that if x is the complement of a generator and (t,n) is in x, then

T(,n)E* {(s, m) E T : (t,n) < (s, m)}E x

(for type 1 generators this is a consequence of the previous fact) . By considering the disjunctive normal form of F and applying this observation the fact follows immediately. ID

2.2 Being Separable

An ultrafilter v in d records a unique subset c of N which splits a portion of the gap (for every n there is a unique tn C n such that T(t,,,) E v - let c = un-,- t,). If this ultrafilter is extended to a Bter in 8 which contains type 1 generators, then the pairs (a, b) corresponding to these generators must be split modulo K. by the set c (note that even though K is not an ideal, this notion still makes sense). The following proposition gives LE our &st indication of how properties of the gap r translate into chain conditions in the Boolean dgebra B.

Proposition 2.2.1. If both A and B are a-directed vnder C* then B is not o-centered (and hence st(B) is not separable).

Proof. Suppose B is the union of countably many ultrafilters {v,)~=,. Then it is possible to find a countable sequence c, of subsets of N which correspond to the unique infinite branch each v, determines in (T, <). For each n pick a pair (an,b,) E A 8 B which is not split by ç, modulo Z (i.e. either a, izc, CHAPTER 2. GLOBAL CHALN CONDITIONS 17 or 13, ,Lx G). Since both sides of the gap are a-directed, fkd a pair (a,b) such that a, C* a E A and b, C* b E B for all n. Notice that we may assume (a,b) is in A 8 B. Pick a m such that T(o,b)is in v,. The set c, splits (a, 6) modulo K and hence splits (a,, b,) modulo Z, a contradiction. O

2.3 Souslin's Condition

Element of the Boolean algebra B can be thought of as a collection of splitters for some portion of the gap J? modulo the compact set K. If we axe given that in A

1. every uncountable C C A contains an uncountable Co which is C* bounded then there is a standard approach to showing that B satisfies Souslui's condition. The general idea is as follows. If F C B is uncountable, then consider the members of A used in the definitions of the members of F. Using 1, refine

F to an uncountable subfady Fofor which there is a co in A which bounds any a' such that T(a~,brlis mentioned in Fo.By making a hite modification to CO, it is possible to produce a c Ç W which splits many members of Fo. The real tri& turns out to be how to make this finite modification.

Loosely speaking, we are given a collection .F of tinite pieces of the gap, where each F E F is split by some "local" splitter CF modulo K. We are &O given some "global7' splitter c which works for all of the pieces in 3, but only modulo the larger object Z. The goal is to repair c by altering some kite portion of it so that it also splits many members of 3 modulo K. CNAPTER 2. GLOBAL CHAIN CONDITIONS 18

The sdficient condition whîch 1 will use is that (A, B) is actudy orthogonal modulo a smalIer ided 3 which satisfies the following "exchange" property:

2. For every J in 3 there are infinitely many n such that for every K E K

the set (J\ [O, n - 11) U (K n [O,n- 11) is in K.

We are now ready to prove the following proposition about B.

Proposition 2.3.1. If A, B, Z and 3 satisfy 1 and 2 then B satisfies Shanin's condition and in particular satisfies Souslin's Condition.

Proof. Pick an uncountable family F of positive elements of B. Applying Fact 2.1.5 it may be assumed without loss of generalifiy that the members of 3 are the meet of finitely many generators. For each F E F pick a finite set SF C r and a (tF,RF) in T such that

Applying 1 it is possible to find an uncountable FoS F and a co E A such that a C* CO whenever a E ra(Si.)and F E Fo.If F E 30,let

and choose a CF such that (cF n n, n) E F for all n. AppIying 2 hda NF > np such that CHAPTER 2. GLOBAL CmCONDITIONS 19

is in X: whenever K is in K. Now pi& an uncountable subset FIof Fosuch that NF = N andcFn[O,N-11 =tforsomefxed N ~NandtC [O,N-l]

whenever F E 31.Let c = (co \ [O, N - 11) ut. I will now show that (cfln,n)E F for all F E FIand n E K Let F E FI

and (A,B) E SF.Since F C S(t,~)n T(=,b) is nonempty,

are both in KN C /C- It follows from the choice of N that

a\c c KaU (JF\ [O,N-11) and bric c Kb u (JF \ [O, N - 11)

are both in K: and therefore (c n n, n) E TCosb)for au n.

2.4 Being Linearly Fibered

The property of being linearly fibered is, not surprisingly, closely comected to whether the gap is linear.

Proposition 2.4.1. If r is well ordered by C* x C* then the map 4 : st(B) + st(A) defined by 4(u) = u 1 A has fibers which are lznear cornpacta.

Notation. In this situation we will let l? = {(a(,bc) : E < K) be an increasing enumeration of r. For simplicity we will write Tc instead of CHAPTER 2. GLOBAL CHAIN CONDITIONS 20

Proof. Let u be an ultrater on A and define r, to be the collection of dl 6 < X for which {TF)U v is a filter. It now suflices to show that if 7 < 6 E r, then T, 1 v 2 TE1 u.

Pick a m E N such that

\[O,m-l] C ac\[O,m-l],

b, \ [O? - 11 C bc \ [OF- l] and let s Ç [O, rn - 1] be the unique set such that T(,,,)E v. If (s,m) c (t,n) is in TEthen

%n[m,n-il Ç aEn[m,n-11,

b,n[m,n-11 Ç b

Because T,nT(,,,) # 0, (%n[O,m-l])\t E Km.Therefore (a,n[O, n-l])\t E

K, andb,n[O,n-l]nt~K,. Thuç (t,n)ET, mdT, [V >TEru. O

2.5 Helly's Condition

lnterval algebras have a very interesting intersection property discovered by

Helly in [32]: a collection of intervals is linked ifE it is centered. Motivated by this I will dehe CHAPTER 2. GLOBAL CHAIN CONDITIOIVS 21

Helly's Condition: There is a base of sets S such that every linked sub collection of S is centered.

Clearly if a a-linked Boolean algebra satisfies Helly's condition then it is actudy o-centered. It tums out that Helly's condition corresponds to a natural consideration in our method of construction as well.

Proposition 2.5.1. If K is an ideal then B satisfies HeZly 's condition.

Remark 2.5.2. Compact ide& are always trivial - they are of the form P(A) for some subset A of N.

Proof. The base in question is the set of generators. First notice that any Iinked collection of generators of the form T(,,,) is centered. With a slight amount of argument it is easy to verify that it is enough to consider only linked collections of the generators T(a,b)-Suppose that F C I' is finite and

(T(a,6): (a, b) E F) is linked. This means that for every pair (a,b), (a',bf) in

F there is a c C N such that a, af CK c and b, b' IK b are in K. Siace K: is closed under hite unions, this implies that (a n 6') U (a' n 6) is in K. Rom this it follows that

has the property that cfn b is in IC whenever (a,b) is in 3.Hence c n [O, n - i] is in T(a,b1 for every n whenever (a, b) is in F. CHAPTER GLOBAL CONDITIONS

For the Grst application of the above construction, we will build a linearly fibered compactum which satisfies Souslin's condition (even Shanin's condition) but which is nonseparable. The set theoretic assumption which we will use in the construction is minor in the sense that it actually follows £rom PFA, an axiom which should identify the chah conditions in as maay compacta as possible. Our assumption was removed in a related construction due to Todortevib of a space with the same properties as ours (see Theorem 8.4 in [66] and the remark which follows) .

The assumption which we will use is Nl < add,(tl) = c. Working in ZFC we will build a pair of analytic P-ide& A and B which form a gap both over their intersection 3 and ako over a larger Z which is an F, P-ideal. The assumption NI < add,(ll) = c will allow us to find cofinal C' chains A. E A and Bo C B, each of which is N1-directed. I' is then chosen appropriately inside of A. @ Bo. It has recently been shown by TodorCeviC (see [69]) t hat if A is an analytic P-ideal then (el, 5) can be mapped monotonicly and cofkally into (A,C).

Here ti is the collection (x E RN : Ix(n)l < oo) of absolutely convergent series and the order 5 is the coordinatewise order. If add,(A) and add,(tl) are defined to be the sizes of the smallest unbounded families in (A,C*) and (4, If)respectively, then it follows that add,(ll) $ add.(A). Thus A and B are NI-directed and contain a cohal C'-chah under our assumptions. 1 will now construct an analytic gap with the properties required for our CRAPTER 2. GLOBAL CWCONDITIONS 23

compact space to be nonseparable, satise Souslui's condition, and be linearly fibered:

1. A and B are analytic P-ide&.

2. .7 and K satisfy condition 2 mentioned in Section 2.3.

3. (A,B) is a gap modulo both Z and 3.

It should be noted that the analytic gap which we will construct below is modeled after a gap of Farah [21] (see Theorem 5.10.2). First it is necessazy to make some preliminary definitions. For a C N define p(a) = xnEA1/n. Let

E, = ln 2 - max{p(a) : a C n and p(a) < ln 2).

Then for all n it is true that 1 > E, 2 E,+I > O. Also it is clear that

E, = O. Define h : N + N by setting h(k) to be the least integer n

such that i/n is less than Define g : + N recursively so that g(1) = h(1) and g(n + 1) = h(g(n)). For convenience 1 will also dehe

hk(n) = h(kn). Let E : [qwt, P denote the canonical bijection which identifies subsets of N with their increasing enumeration. It wiil be useN to think of E as being defined on the on the fbite sets as well: if a set a has no

nth element then set E (a)(n)= m. 1 will use [m,n] denote the interval of integers between (and including) m and n. Rather than working inside of P(N) it will be easier to carry out the construction inside of P(R)where R is defined as follows: CHAPTER 2. GLOBAL CHAIN CONDITIONS

Here a takes the place of [O, n - 11 in the construction. Note that h(n) is at least Sn and therefore

Define the following:

4. Lais the collection of all L C N such that p({n E N : unnL # 0)) < oo

and hk <* E(L) for all k.

6. A= {a CR:r1(a) E Lo)

10. Ic = {K C R : p(r1(K)) 5 ln 2 and p(7r2(K)) 5 ln 2)

Remark. Notice that C L1since whenever 2n <* E(L) it always follows that p(L) < oo. Rom this it is immediate that 3 & 1. Since 3 = An B, it follows automatically that A IJ B and A lz B. CHAPTER 2. GLOBAL CWCONDITIOArS 25

Lemma 2.6.1. Al1 of the collections rnentioned in 4-10 are dense analytic P-ideals and K: is compact.

Proof- The compactness of K: follows fiom the fact that for any set L Ç N, p(L) > ln2 ifF p(F) > ln2 for some fite subset F of L. It is a routine exercise in descriptive set theory to ver* that al1 the remahhg objects are anaIytic- It is easily seen that Clis a dense P-ideal and since TI,7r2 are finite-to-one maps, it suffices to show that

is a dense P-ideal. Let (Lk)& be a sequence of elements of L. For each k, pi& a nk > k such that hkck+li(n) < E(Li)(n)whenever i < k and n > nk. Now let 00

L = u Lk \ [li nk]. k=l To see that L is in C,let k E N be given and q > nk and r < k. Notice that

Furthermore the right hand side is at least

by our choice of q (note that r < k 5 nk < q). The density of L follows fkom the fact that for any f E TVN and any infinite set L Ç W, there is an infinite set Lo such that f <* E(Lo). Cl CHAPTER 2. GLOBAL CHARV CONDITIONS

Lemma 2.6.2. If J is in 3, there are Znfinitely many n such that

for i = 1,2 and hence 3 and K satisfy condition 2 of Section 2.3.

Proof. Pi& a No such that h(n) < E(ri(J))(n) for all n > No, i = 1,2 and let N = ma- E(ri(J)(No+ 1)). Notice that for all n, i = 1,2

Now for infinitely many n > N, un x un flJ = 0. Thus for such n

The proceeding inequa.lity follows hom this dominance and the fact that the least element in ri(J \ &) is at least g(n + 2) = h(g(n+ l)),for i = 1,2. O

Findy 1 will use a Fubini style argument to show that (A,B) is indeed a gap as 1have prornised all along.

Lemma 2.6.3. (A,B) is a gap in P(R)/T-

Proof. Suppose that c E R. If a C R,define CHAPTER 2. GLOBAL CHXlN CO.iiITIONS 27 and note that this is just the product measure when restricted to the hite rectangles un x (since p is determined by its value on the singletons). Set

1 now will consider two overlapping cases.

Case 1. L = (k E N : rk 2 112) is infinite. Let LI be an infinite subset of ~5such that p(&) is hite. Then for each k in LI,let nk be an element of

~k SUC^ th&

~({m: (m, nk) E 4) 2 (1/2M~k)-

This choice is possible by Fùbini's theorem. Now let

By choice of nk,

On the other hand, ?rz (b) n a, contains at most one element and thus hi, <* g 5, E(744) for dl k. Furthermore {n E N : 7~2(b)n un # 0) = LI and therefore b E B \ 55 Case 2. N\ L is infinite. It is now possible to apply a symmetric argument to find an a E A\Zsuch that OC=0.

2.7 An Example From b > N2

It should be clear that the previous example fails to satisk Helly's condition. As the example of TodorEeviC in Theorem 8.4 [66] is clearly dinked (and CHAPTER 2. GLOBAL CHAIN CONDITrONS 28

hence fails to satis Helly's condition as well), it is natural to ask whether such spaces can be irnproved to satisfy Helly's condition. It tums out that, to some extent, this is not so. The compatibility relation between the generators in these algebras is clearly closed (when viewed as a subset of [ïl2)and hence is subject to the consequences of OCA.

OCA : If X is a separable metric space and G C [XI2 is open (with [XI2 viewed as an appropriate subset of X2) then either G is countably chromatic or else G contains an uncountable complete subgraph'.

This axiom implies that all such Boolean algebras which satisfy Souslin's condition are O-linked. Thus, if there is an example of a linearly fibered compactum which satisfies Souslin's condition but does not have a a-linked base, it either must require a set theoretic assumption which is incompatible with OCA or else must differ considerably from the examples rnentioned in this chapter. Concerning this, 1 will reproduce (in the presence of the machinery of this chapter) a construction of TodorEevii: which appeared in an early draft of [66]. The assumption b > N2 implies that there is a gap (A, B) with the following properties (see Lemma 3.10 of [74]):

'This is the formulation of OCA currently quoted today. Earlier versions of this axiom for sets of size N1 can be found in [Il. The only aspect of OCA which we wiIl use in this thesis is its ability to ('reflect:" if G isn't countably chromatic then it contains a subgraph of size NI which is not countably chromatic. GLOBAL cmCONDITIONS

2. B is countabiy directed.

3. A and B are well ordered by C*-

4. A and B are orthogonal modulo finite.

5. (A,B) is a gap modulo hite.

Here Z = 3 = fin and K: consists only of the empty set. It follows that the Boolean algebra D associated with this gap produces a compactum which is nonseparable, satisfies both Souslin's and HeUy's condition, and is linearly fibered. Such an algebra also exists if b = NI (the example is, moreover, metrizably fibered), though the construction of this example does not involve the use of a gap (see [80] and Section 3 of [66]). Chapter 3

Analysis of Local Chain Condit ions

MAN,was an axiom which came out of the solution of Souslin's problem - Martin realized that the techniques which Solovay and Tennenbaum used to produce a consistent positive answer to Souslin's problem in [57] also gave the consistency of the following statement:

MAN,: A compact space which satisfies Souslin's condition can not be covered by NI nowhere dense sets.

Even before it appeared in literature in [57], this axiorn was already shown to have a number of consequences (see [22]) :

1. Souslin's condition is equivalent to Shanin's condition.

2. Souslin's condition is productive. CNAPTER 3. LOCAL CHjURr CONDITIONS

3- The union of NI measure O sets has measure O.

4. Every tree of size NI in which all branches are countable is the union of countably many antichains.

5. There is a non-fiee group A such that whenever T : B + A is a homomorphism with kernel Z, there is a homomorphism p : A -+ B such that TP is the identity (there is a W-group which is not free).

While MAH,is, on the surface, a statement about Baire category, it was quickly realized that this axïom has a strong idluence on the identification of chah conditions in topological spaces (see 1, 2, and 4 above). That the identification of chah conditions in turn has a strong influence on Baire category, however, came as somewhat of a surprise.

Theorem 3.0.1. [83] MAHlis equiwzlent to identifying the Souslin's condz- tion and Shanin's condition in the class of compact topological spaces.

In light of this, it became quite natural to ask whether Shanin's condition could be relaxed to some weaker local chab condition. Research in this direction so far has been focused on establishing which consequences of MANl follow from identimg, for example, Souslin's condition and Knaster's condition (see [67], [74],and [78]). To help facilitate the discussion, define the following axioms, for n 2 2:

?Cn: Every compact topological space which satisfies Souslin's condition satisfies Knaster's condition K,. CHAPTER 3. LOCAL C.CONDITIONS 32

Here Knaster's condition K, asserts that every uncountable collection contains an uncountable n-linked subcollection. The focus of this chapter will be to remark on some specific consequences of IC2 and K3.Typically a discussion of this sort is carried out in the language of partitions (see [67] and [74]). We will present the fht section in the language of partitions and the latter in the language of Boolean dgebras for the sake of contra&

3.1 Some Consequences Kz

In [78] TodorCeviC dernonstrated that ICz implies that every tree of size NI without an uncountable brmch is the union of countably many antichaùis- The following is a more generd form of this result which was also proven and stated in [78]. 1 have included a sketch of the proof for completeness and since it involved an interesting new technique.

Theorem 3.1.1. [78] (&) If G C [wlI2 is a graph then either G is countably chromatic or else there is a pair of sepuences De, EE (E < wl) of disjoint n

elernent sets such that for every E # 71 < WI DE x EEn G is nonempty.

Before proving this result, let us hrst examine some of its consequences. If T is a tree on wl without an uncountable branch then the compatibility graph G on T can never satisfy the second alternative of the above theorem (see Section 9 of [82]). Also, it should be clear that "T can be decomposed into countably many antichains" is just a reformulation of "the compatibility graph is countably chromatic." Thus we have the following. CHAPTER 3. LOCAL CWCONDITIONS 33

Theorem 3.1.2. [781(K2)If T is a tree of size NI and T has no uncountable branch then T can be decomposed into countably many antichains.

&O, iC2 has an influence on the measure of small sets of reals.

Theorem 3.1.3. (&) Every set of reals of sire N1 has measure O.

The reason for this is as foIlows. Tail [62] has shown that a set X Pis a- discrete in the density topology ifT it is closed discrete in the density topology iff it has measure O. Subsets of R of size NI are always left separated in the density topology. Thus if xe (< < wl) is a sequence of distinct reals then there is a sequence of closed sets of positive measure E, (< < wl) which contain x, as a density 1 element but which do not contain any xc for < q. If G is detined by putting CE, 7) in G iff x, is in then G can not satisfy the latter condition of the above theorem (see the Claim on page 62 of [74]). If l& holds then G is countably chromatic. Thus X = {xc : E < wl) is a-discrete in the density topology and hence measure O. Now we shall sketch the proof of the theorem.

Proof. Fix a sequence of 1-1 maps e, : a! + w for each a < wl such that e, e, =* ep r a whenever cr < p < wl (such a sequence can indeed be found - see [73] or [?, Ch. 41). Set r, to be the range of e, and defke {a,P) E K iE for all n < A (r,, rg j such that n E r, (equivalently rg) {e~l(n) , eil(n)} is in G.

By the remarks made in Section 1.3 K2 implies that if K is C.C.C. then there is an uncountable H E wl such that [Hl2 C K. If such an H exists, CHAPTER 3, LOCAL CHALlV CONDITIONS

defhe for s G {O,. . . ,n) by

It is easily checked that the A(s,n)'~cover wl and each satisfy [A(s,nlj2fi G is empty. This G is countably chromatic. Now suppose K is not c-c-c.. That is there is a sequence FE (E < wl) of disjoint n element K homogeneous sets such that FE x F, is not K homoge- necus for any 5 # 7. For each 5 < wl pick a pair of distinct ordinals 4, EF greater than 6 and a integers m~,ne such that:

2. ea r E = ep r E whenever cr is the ithelement of Fs, and ,B is the ith element of F,, (ithin the order on wl).

3. A(r,,rp) < rn~whenever a is the ithelement of FQ and ,û is the jth element of F,, for i # j (ithand jthin the order on wl).

4. me < A(ra, rat) < nEwhenever o: is the ithelement of Fs, and a' is the ithelement of FE,.

Define

DE= u{~;'{o,. . . ,ne): a E Fs,)

EE = ~{ep'{~,... ,ne) : a E F,,).

Rom item 2 it follows that DEn 6 = EEn E. By pressing dom and rehing to an uncountable set r & wl we may assume that: CWTER3. LOCAL CMCONDITIONS

6. A(T,,rat) > n if cu is the ithelement of DE (respectively Ec) and a' is the ithelement of D, (respectively E,).

7. if m 5 n and e&) = m where a is the ithelement of DE for E in l?

then e&) = m where /3 is the ithelement of E, for any 7 in î.

8. if < 7 then -q is greater than the maximum of De U Be.

We are now finished once we prove the following claim.

Claim 3.1.4. If (DE\ E) x (E, \ q) n G is ernpty then F6c U F, is K homogeneous.

Proof. Assume that (De\ E) x (Eq\ 7)fi G is empty. Suppose that a is in Fs, and fl is in F,. By 3,4,and 6 we know that A(r,, rp)< n. Now suppose that cl and 6 axe such that e&) = es(&) < n. Notice that if Ç1 or C is less than E or 0 respectively then G = by 7. Thus Ci > E, C2 > 17 and by om assumption {Ci,6) is not in G and hence {a,,8) is in K and Fs, U F., is homogeneous. O

3.2 IC3 and Additivity of Measure

The focus of this section will be the following result . The key ideal was already present in the proof of observation (15) in [78]. The general machînery which we will use in our construction was developed in [80]. CHAPTER 3. LOCAL CILAIN CONDITIONS 36

Theorem 3.2.1. There is a 0-linked Boolean algebra D such that, if there is a family of & measure O sets whose union is not measure 0, B does not satisfjr Knaster's condition K3.

Notation. Z denotes the collection of a.U fite unions of rational intervals.

We will need the following standard fact about measure O sets (we will prove a variation of this in Section 4.1).

Fact 3.2.2. For euery measure O set G IR, there is a sequence FG : N -+ Z such that the measure of Fc(i) is ut most 2-2' and

00 00

For each measure O set G, we will fix a sequence FG as above once and for A.

Notation. T denotes di (t, n) such that

1. nis inN

2. t is a subset of Z"

3. t (viewed as a finite tree) has no node with more than two immediate successors.

Remark 3-23If (t, n) is in T then condition 3 implies t has at most Zn many elementS.

For a measure O set G, define Tc to be all (t,n) such that the restriction of Fo to [O, n - 11 is in t. Define B to be the subalgebra of P (T)/fb generated by TG for G measure O. CEFAPTER 3. LOCAL CRAiN CONDITIONS

Claim 3.2.4. XI is cr-linked.

Proof. For A in B there is a sequence of rneasure O sets Gi (i < 1) such that

Define n~ to be the maximum of all

for 1 5 i # j 5 k. If A is positive then there is a ta such that (ta,na) in A. The collection of all such A with (t,n) = (ta,na) is Iùiked. O

Claim 3.2.5. If {TG : G E E) is 3-linked then the union of E has measure o.

Remark 3.2.6. Notice that this fkishes the proof of the theorem. If F is a fdyof measure O sets of size NI whose union is not measure O then, by taking some unions, there is a collection E of measure O sets such that

U3= uE' = UE for every uncountable E' 2 E. In this case (Tc : G E E} will contain no uncountable 3-linked family.

measure at most Si .-2-2i = 2-i. Notice that for ail k CHAPTER 3. LOCAL CHARV CONDITIONS and

Thus the union of E has measure O. Chapter 4

Cont inuous Irreducible Colorings

In this chapter, we will study strong failures of Ramsey's theorem at certain cardinds associated with the continuum. A coloring of pairs of a set X is simply a map fiom the set [XIZof ail unordered pairs of elements of X into some set Z of colors. A c010nng c : [XI2 Z is said to be irreducible if for every Y C X of the same cmclinality as X, d'[YI2= 2. For arbitrary colorings using 2 colors, Sierpiiiski (in effect) gave a defi- tive counterexample at the Ievel of the continuum.

Exampie 4.0.7. [52] There is a coloring c : [RI2-+ {O, 1) such that c is not constant on any set of pairs of an uncountable set.

The coloring c mentioned above makes crucial use of a well ordering of the continuum - in fact the value of c on a pair is determined by whether or not a fixed well order on R and the usual order agree. It is therefore natural to ask for irreducible colorings which are more well behaved in some sense. One natural restriction to place on the colorings is to ask that they be continuous. Here X and Z are assumed to have a separable metric topology associated with them. In our case Z will typically be countable and discrete. Since P is the universal PoIkh space, there is no loss of generality in considering only those X which are subspaces of P. In this context, the continuity of c is equivalent to the property that ody a hite number of entries in x and y are needed to compute c(x,y), thus making this a very natural restriction indeed. The general question is to understand when it is possible to find continuous colorings on sets of reals of a certain size. This is perhaps a little misleading, since in hindsight it is more how a set of reals behaves in a certain structure which is interesting here, rather than its cardinality. For example, we will see that there is always a continuous irreducible coloring on a set of realç of size 6, the cardinality of the smallest unbounded subset of (P,<*)

where P is the collection of all increasing functions £rom ~c;to W. The theorem at hand actually tells us that there is a continuous irreducible coloring on any unbounded chah in (P,<*) (it is well known that there is always an unbounded chah of size 6). Thus cardinal invaiants are simply a convenient language for discussing results about continuous irreducible colorings. This chapter has three sections. Section 4.1 investigates a coloring which is closely connected to the ideal of Lebesgue mesure O sets. Ironically it makes use of a lower bound on a class of finite Ramçey numbers. Ideas from this section were already put to use in Section 3. Section 4.2 revisits the oscillation map of TodorCeviC and introduces a variation cded the alternation map which is continuous. Section 4.3 puts this mapping to use to produce a new proof that PFA implies c is N2. The ideas presented here may be relevant to the problem of whether or not OCA implies that c is N2.

4.1 A Coloring Associated With Measure

In this section we will study a subfdy of the Lebesgue measure O subsets of R. This family is natural to consider in the sense that it consists of those measure O sets which are measure O for a concrete reason and in particular encompass all of the measure O sets which one is likely to encounter outside of a set theoretic discussion. The family is of interest since the property of being unbounded in the ideal can be co~ectedto continuous irreducible colorings. To define the family of interest it is first necessary to make a few defbi- tions. Suppose that G C 2" is a Gg set of measuce O and that G = Un where Un is an open set of measure less than 2-". For each n, let {[tz]}be a List of disjoint clopen sets whose union is Un.Let F(i) be the collection of al1 t$ in 2" NOWnotice that Un C Uzn[F(i)]since U,has measure less than 2-"* Also, Ca

Thus a set G 2" has measure O iff there is a sequence FG = (FG(I): i E w) such that CHAPTER 4. CONTWOUS ZRREDUCBLE COLORLNGS

Such a sequence will be called a cover of G. If r E (0, l),a cover FG of G is

Thus nice covers are those for which the sum mentioned above converges for a speciûc reason - the same reason that Cr=,l/nP converges whenever p > 1. Dehe NF to be the collection of all subsets of 2" which have a r nice cover. Notice that if r < s then NT Ns E N. If d C B are families of measure O sets, define add(A) to be the size of the smallest subset of d which is unbounded in (A, C) and add(A, B) to be the size of the srnallest subset of A which is unbounded in (8,Ç). It is easy to see that add(N,, Ns)5 non(Ns) $ non(N). Also, if r 5 a < b 5 s then add(N,, Nb)5 add(Nr, Ns).Thus using a weU foundedness axgument on the ordinals, it is possible to hda pair r < s in (O, 1) such that for all a < b in [r, s], add(N,, Ns) = add(N,, Nb).Dehe

= n,,,Nb.

Lemma 4.1.1. The invariants add(N*) and add(N,, N,)are equal. More- ouer there are a < b in the interual (r, s) and A C Na of sire add(N,) which is well ordered by C and unbounded in Nb.

Proof. It is easy to see that N, E N. & Ns and therefore that add(Nr, Ns)2 add(N.) . Now let A be an unbounded subset of N. of size add(N.) . It can be assumed vrithout loss of generality that A is well ordered by G. Since A is unbounded, there is a b > r such that UAis not in Nb. Let a be any element of (r,6). Now clearly A c Naand is unbounded in Nb. Thus A must have size add(N,, Nb)= add(N,, Ns). O

From th% point on a and b will remain £ked and d will be any number such that a < d < b.

Theorem 4.1.2. There is a subset X of wu of size add(N.) and a continuous

irreducible colorhg c : [XI2 + W.

The existence of the desired irreducible partition for the cardinal add(x) is a consequence of the following sequence of resdts. It is perhaps surprising that this coloring makes crucial use of exponential lower bounds for a certain class of finite Ramsey numbers. The following fact for k = 2 and 1 = 1 is essentially a well known result of Erdos [19](see also Section 26 of [20]) ' and the methods presented in the proof can readily be adapted to give us the following result.

Theorem 4.1.3. If (f) 5 rn! then (m)&> (k/l)(m-1)/2.In partzcular there is a constant cu > O such that [ml; > 2am/k.

Proof- First notice that for a fixed integer n, the number of colorings c :

[nI2+ k is N = k(i). If S is a fixed subset of n of size rn and L Ç k is a collection of 1 colors, then there are l(l)k(l)-(;) many colorings c : [nI2-t k

'Erdos actually showed (m)$l > 2m/2. CEAPTER 4. CONTINUOUS IRRED UCLBLE COLORINGS 44 which aIso satisfy d'[SI2C L. Since there are (m) ways to choose S and (:) ways to choose L, there are at most

many colorings c of [nI2 such that there is a subset of n of size m which realizes at most Z colors. It now suffices to show that if n 5 (k/~)(~-')/~then NmVl< N. I mill use the approxïmatioii (m) < nm/m!.

To see that the constant a exists, note that

Suppose that T is a subset of ww and f E ww is a function. I will say that T is f thzn if for all but finitely many n in w and every t in wn the concatenation t ^i is a node of T for at most f (n) many i (Le. the splitting of the nodes of T is bounded by f). Sidarly 1 will cdT f splitting if T does not contain a f thin subset of the same size. Define h(n) = 2J" and let hk be the k fold composition of h with itself. Remark: 4.1.4- F'reguently we will be interested in taking the integral part of a real valued function. 1 will sïmply write the reai valued function and let it be understood that what 1redy mean is the greatest integer less than this function.

Lemma 4.1.5. Suppose T is any subset of wW zuhich is h( f) = 2fi thin for some f with lim, f (n) = m. It follows that there is a continuous coloring c : [Tl2+ w such that d'[TOI2= w for any ToÇ T which Zs not f thin.

Proof. Notice first that for any h( f)thin subset T of wu there is an isometry which embeds T into nz=,h( f (n)) (this is in fact an equivalent formulation of "thinness" ). Thus 1will work in nr=,h( f (n))for convenience. Let G(n) be the greatest integer k such that

2"f(n)'k < h(f (n)).

Note that G is both well defined and satisfies lua, G(n)= m. For each n

such t hat for every subset S of h(f (n)) having size f (n), C: [s]= G(n) (this is precisely what the definition of G(n)and Theorem 4.1.3 guaranteed). Now

where n = A(x, y). To see that c has the desired properties, let k E w be arbitraq and To T be as in the statement of the theorem. Pick a m such that G(m)> k. Now End a n 2 m and a t in wn such that the set

S = {i E h(f(n)) : 3x E S(tniC x)} has at le& f (n) elements in it. Then c:[sl2 = G(n) contains k and is contained in c"[T~]~by definition. Cl

Notice that if T C wu is a hk(f)thin set which is f splitting then Lemma

4.1.5 tells us that there is a continuous irreducible coloring c : &I2 + w for some To C T of the same size. To see this, set Sk = T. I£ Sk is hk-~(f) splitting then this is a consequence of the lemma. If not let Sk-l be a hk-~(f) thin subset of Si, having the same cardinality as Sk. Now try to apply Lemma 4.1.5 to Sk-l and so on. For some i >_ 1, Sihas the same size as T and is hi(f ) thin but hiei (f ) splitting (otherwise this would contradict the fact that T is f = ho(f) splitting).

Lemma 4.1.6. Suppose that T C ww is 22n thin but there is not a To C T of the same size such that #(To 1 n) 5 nd-afor al1 n. Then there is a sabset Z of ww with the sarne cardznality as T and a continuous irreducible colom'ng of Pl2-

Proof. For each x in T dehe z, (n) to be the finite sequence

Notice that since T is 22n thin, Z = {z, : x E T}is (22'n+1'2)(n")2 thin.

It is easy to verZy that for some k, hi(nd-") 2 (22'n+1'2)(n+1)2for a.U n. It therefore sufEces to show that Z is nd+ splitting. Suppose that Zo is a nd-athin subset of Z of the same size. By refining Zo if necessary it may be assumes that nd-acontrols the splitting at all nodes of Zo. Let To= {x E T : r, E Zo) and note that To has the same size as T. 1 wilI now prove by induction that #(To r n) 5 nd-a for all n. Notice that, if necessary, it is possible to pass to an appropriate neighborhood of To to ensure that the base case of the induction is satisfied- Now suppose that #(To 1 m) 5 md-a for all m less than n. Fix a m such that m2 5 n < (m + I)~.Then #(To m) 5 md-aand for each t in To 1 m there axe at most md-a many t' in To r (rn + 1)2 which extend t. This latter fact follows hom our assumption that Zo is nd-athin. Thus To r n can have at most

many members, a contradiction. O

The following is the last lemma which is needed to show that there is an irreducible coloring associat ed with add(N. ) .

Lemma 4.1.7. There is a 22n thin subset T of wu of cardinality add(N.) uihich has no subset To of the same sire satisfying #(To 1 n) 5 nd-afor al1 n.

Proof. Fix a family A C Na of size add(N.) which is well ordered by C and unbounded in Nb. For each A in A choose an a nice cover of A which also satisfies FA(n)n2/2"5 na for all n. Let T = {FA : A E A} and notice that T is 22n thin. CHAPTER 4. CONTLNUOUS LRREDUCIBLE COLORINGS 48

Suppose for contradiction that there is a To T with the same size as T and which satisfies #(To r n) < nd-a for al n. Define &(n) = u{FA(~): FA E Ta)and notice that

Thus #(Fc(n))n2-d/2n 5 1 for all n. Since mnd/nb= O, FG is a b nice cover for u{A E A : FA E TO).If TOhas the same size as T, then

a contradiction.

4.2 The Alternation Map

In [72] TodorFeviC introduces and studies the behaviors of the oscillation map (see ako Chapter 1 of [74]). If x and y are two elements of P, the set of all strictly increasing functions, then osc(x, y) is defined to be the number of times x and y "oscillate." More formdy osc(x, y) is the minimum size of a collection of intervals Z such that Z covers w and whenever 1 is in Z and m, n are in 1 then

As simple as this definition is, TodorEeviE shows that it exhibits a rather remarkable behavior .

Theorem 4.2.1. [74/ Whenever X is an zmbounded and o-directed subset of (P,<*) and n is in w, there is a pair x,y in X such that osc(x, y) = n. Thus if X is an unbounded chah in (IF', <*) which is well ordered in a regular order trpe then osc is irreducible when restricted to [XI2. It is clear, however, that the oscillation rnap is not continuous. The purpose of this section is to produce a continuous variation of the oscillation rnap of TodorëeviC. We wïll focus on the following definition.

Definition 4.2.2. If S w and x and y are in P then x and y alternate on

S if for every consecutive pair m, n of elements of S

This definition also makes sense if x and y are only partial maps, so long as they are both defined on S. Define S : [Pl2+ P is defined recursively as follows. If x <,, y are in P, then dehe S(x,y) w recursively as follows. The first element of S(x,y) is

A (2,y). Given given the 2nth element S(x, y)(2n) of S(x,y), dehe

Notice that S : [Pl2 i [wlW is a continuous map. Also, S can be defined for pairs of fite sequences as well. In this case, the procedure for choosing members of S(s,t) stops when the prospective member of S is no longer in the intersection of the domains of s and t. If x and y are in P then define alt(x, y) to be the size of the largest initial segment of S(x, y) on which x and y alternate. Since S is continuous, alt is continuous as well. CWTER4. CONTLNUOUS IRREDUCLBLE COLORllVGS 50

It turns out that the map alt exhibits many of the behaviors of the map osc as the foUowing theorem demonstrates. As it will be needed in the following section, 1 will prove the next theorem in its full strength (which immediately gives a continuous irreducible coloring in the case k = 1 for a set X unbounded and well ordered in (P,<*)). Before stating and proving the theorem it will be useful to first defme some notation and terminology.

Notation. If x is in [xIkwhere X P is totally ordered by <* then xj is the jthle& element in (x, <* ) where O 5 j < k. If t is in [w'] and the values elements of t at 1 - 1 are distinct then sj is the jthelement of t ordered by the last coordinate.

Notation. If x and y are in [XIk and [XI'respectively then x <* y abbreviates xj <* ~j'for ail j < k and j' < 1 (similarly one defhes x <" y).

Definition 4.2.3. If X 5 P then F C [XIk is cofinal if for every x in X, there is a y in F such that x <* y.

Definition 4.2.4. If F C [XIk for some X C P and t E [wnIk then t is a splitting node of F if for infinitely many i there is a x in F extending t such that xj(lt1) > i for aJ O 5 j < k.

Theorem 4.2.5. If X is well ordered and unbounded in (X, <*) and F C [XIk is cofinal then there is an integer 1 and a sequence x; (i E w) such that for every i and every j 5 k 2. 4-(1)= 4i2(1) for all i and O 5 j < k, and

Proof. It may be assumed without loss of generality that there is a t-1 in [w'Ik such that

1. Ll is an initial segment of every x in F,

2. d <' d' whenever x is in F and O 5 j < j' < k, and

3. is a splitting node of F.

Let D C F be a countable dense subset and a E X be a <* bound for D.

Pick a ndland refbe F to a cohal set E such that a <"-1 x for every x in Mk E. Recursively construct for i in w integers ni, elements ti of UM=o[~] , and elements of D such that:

1. Llis an initial segment of ti for i 2 O and ~(4,<+2m+l) = 1 = Z fora.llO

2, ti is a splitting node of F. Carrying out the recursion is routine. Notice that ?$ and $+, alternate on ~(e,e+,) n {O, . . . , ltil). This follows from 7. Furthermore, by 5, 7, and 8 alt(G, si+,) = i. Since q+lextends si+l, z$+~)= i. The other two conditions we are interested in follow immediately from 2 and 6. O

4.3 Continuous Codings for Real Numbers

The focus for this section is to use the map c from the previous section to code red numbers using partitions. The purpose of this is twofold. First, it generates a new proof that PFA implies c is b (and hence N2)2. Second, it gives an example of a coding device whïch is continuous and therefore subject to the consequences of OCA. Methis still leaves open the question of whether OCA in fact implies that c is Na, it does seem to shed some new light on this problem. The crucial objects of our study will be the following binary colorings

: [xI2+ {O, 1) defined for each real r in 2W. Let 4 : w -+ 2

( O otherwise.

The maps ç are continuous since c is continuous. Also, no infinite set N can be 1-homogeneous for both c, and c, for two difFerent r and s in 2" (H is

*The original proof is due to TodorCeviC and can be found sketched in [68] and in more detail in [7]and in [86]. 1-homogeneous for ç if c is constantly 1 on [Hl2).TO see this, note that by Konig's lemma, either N contains a sequence which converges in B or else

there is an infinite subset Ho of H and an n such that A(+, y) = n for all x, y in Ho Now it is possible to choose a pai~x

For a <*-increasing sequence X = {x, : a < 6) C P of regular cardinal

length define the following statement :

(Sx)For every r in 2W there is a 6, < n of uncountable cohality and a se quence a, of subsets of 6, such that {x, : a E Er,) is 1-homogeneous for c, for all n and UzoEr, contains a closed unbounded subset of 6,.

Notice that (Sx)implies c is at most 6 and at least N2. We WUnow see that PFA irnplies (Sx)for all unbounded <*-increasing sequences X C (IF', <*) and hence PFA implies b = c. Let X be a fixed unbounded <*-increasing sequence in (P, <*). If Y C X then define Q(&, Y) to be the collection of all finite subsets of Y which are 1-homogeneous for c,.. Observe that if Y is unbounded and well ordered in

type wl by C*then every finite power of &(ç, Y) satisfies the countable chah condition. This is an immediate consequence of Theorem 4.2.5. Now let P be the standard partial order for collapsing Ire( to NI without adding reals.

In V', there is a closed unbounded set È in rc of order type wl. Since no new reals have been added, Y = {x, : a f l?) is unbounded in P-Applying PFA to P * QcW(~,Y)fields the objects 6, and {E,,)~=,. For completeness 1 will now add the proof that OCA (and hence PFA) imply b 5 N2 (see Lemma 3.10 and Theorem 8.6 of [74]). If b > Nz, then let

a, (a< w2)be any Ç* chah in [w]" (such a chain of length b exists; just take an initial segment of it). Now, by Zorn's lemma, select another increasing

sequence b, (a< K) such that K is a regular ca.rdina.1, a, n b4 is finite for aJl a < wz and ,û < K, and there does not exist a c w such that a, C* c for a.U a < ~2 and bp n c is finite for all ,û < K. A result of Hausdorff [31] and

Rothberger [51] implies that r; is uncountable since otherwise there would be

an unbounded set of size N2 in (P,<*) .).

Let X be the collection of d (a,6) such that a is disjoint fkom b and a = a,, b = b, for some a < wz and 0 < K. Define G E [XI2by {(a,b), (c, d)) E G iff

(an d) u (b n C) # 0

Notice that G is open. Now if H C X is such that [Hl2n G is empty, then H does not form a gap suice it is split by

It follows from the cardinality considerations on K that X can not be split into countably many collections, none of which form a gap. Hence G is not countably chromatic.

On the other hand, if Xo C X has size Ni, pi& an a < w2 such that a C* a, whenever (a,b) is in X. Let C be the collection of all c whkh differ fkom a, on a finite set. Then CHAPTER 4. CONTNUOUS LRRED UCBLE COLORINGS 55 satisfies [H,I2 nG = 0 for a.lI c in C and hence G 1 Xo is countably chromatic. Thus OCA must fail. Chapter 5

Measure Algebras, Forcing, and MartinSsAxiom

The next three chapters focus on an analysis of random graphs. For us, a random graph is a map G : [SI + R where S is some set (often wl ) and R is a measure algebra. Tt should be noted that this definition of "random graph" difEers fiom the one usually considered in combinatorics (see, e-g., [lG]) and also should not be confused with the random graph on w considered by Rado. Tt turns out that these graphs have applications in both topology and in the combinatorics of wl. Results concerning random graphs fall into two categories. The &st are those which illustrate that some random graphs can exhibit rather unpredictable (or "anti-Ramsey") behavior. These results are generally results in ZFC and usually take the flavor of what staternents always hold concerning graphs on wl after forcing with a rneasure algebra. The second class of results are the dichotomies for random graphs. These require some additional set theoretic assumptions to carry out the andysis and the axiom which has been used for thiç purpose to date is MAN,.While the results are somewhat mied, the ones in the second category generally share the use of a number of common tools and techniques. The purpose of this chapter is to isolate two of these tools and give an introduction to their roie in proving dichotomies.

5.1 Measure Algebras

Before proceeding, let us &st define some terms.

Definition 5.1.1. A rneasure algebra is a pair (R,p) where R is a complete

Boolean algebra and p : 'R + [O, 11 satisfies

1. p(a) 2 O with equaliw holding only for a = 0.

3. p(VzEoai) = CEop(~) whenever ai A aj = O for all i # j.

This is what is usually referred to as a probability algebra (p is usually dowed real or even extended real values). The reader is referred to [23] for a more complete account of the different classes of measure algebras.

Definition 5.1.2. If (R,p) is a measure algebra, then its character r(R)is the cardinality of the smallest subcollection which completely generates it. If the character of R is countable then R is said to be separable. CHAPTER 5. MEASURE ALGEBRAS 58

Definition 5.1.3. A factor 72, of a measure dgebra (R,p) is the restriction of the measure algebra to one of its positive elements a-

A measure algebra is homogeneous if ail of its factors have the same character. Also, 1 will mite the character of (S,v) is eveqwhere greater than the character of (R,p) if the character of each factor of S is greater than the character of every factor of R. The following construction is standard and extremely useful in the study of random graphs.

Definition 5.1.4. If (R,p) is a measure algebra, I is a set, and U is an ultrafilter on 1,then the ultrapower (7Zr/Li,,u) is defhed by setting

taking the quotient of R' modulo the p-measure O sets N, and taking the (unique) completion R1/N with respect to p.

Notice that the map which sends a to the constant map a embeds any measure algebra (R,p) into an ultrapower (R'/u,p) in a measure preserving fas hion. Finally, if R is a complete subalgebra of a measure algebra (S,p) then the projection map .rrz : S -+ R is defined by mR(a) = ~{bE R : a $ b). A useful fact which is easily verified is that if S is the ultrapower of R by some ultrdter U then CHAPTER 5. MEASURE ALGEBRAS 59

The right hand side is in fact the infimum taken over all representatives f' in [fl-

5.2 Martin's Axiom and Random Graphs.

The first essential advance towards understanding random graphs was made by Laver in [41]. He proved that, if MAu, holds, then in VR,any tree of size NI without an uncountable branch can be decomposed into countably many antichains. There were two parts to his argument. The hrst was to show that MAN,impIies that a certain dichotomy holds for all random graphs on wl. The second part of the proof was an interpretation of this dichotomy in the special case consisting of the incompatibility random graph for a tree in VR.Much of the latter portion of his argument can be extracted fIom the next section. The dichotomy result is the following.

Theorem 5.2.1. (MAu,) If G : [wiI2 + R is a random graph over any measure algebra then

1. either there is a sequence X, : wl -t R such that for al1 a < p < wl

00 VX+) =l and G(<~,B)Ax,(~)AX,(P)=O n=O 2. or there is an uncountable sequence Fc (5 < ml) of disjoint n element subsets of wl and a 6 > O such that for every 5 # q Remark 5-22. Notice that the first conclusion is equivalent to the statement that G,when viewed as an R-name for a subset of [ulj2,is forced to be countably chromatic. Analyzing the latter conclusion is dways where new ideas are required when one tries to use this theorem to attack an unsolved problem in this area.

Remark 5.2.3. It should be remarked here that it is only in hindsight that this dichotomy was isolated as a recurring element of many of these arguments and, as such, did not appear explicitly in either [41]or [75].

Proof. Define a partial order (Q,5) fiom G as follows. Let Q be the set of all maps p such that:

1. The domain of p is a hite subset of wl.

2. The range of p is contained in R.

3. For aJl7 in the range of p, p(y) has measure greater than 1/2 and is in

the span of {~(a,,O) : a < B < T).

4. For aJl a < in the domain of p, G(Q) ~p(a)/\p(B) = 0.

If p and q are two conditions in Q then q 5 p iff the domain of q contains the domain of p and q(a) 5 p(a) whenever a is in the domain of p. Notice that, by considering only those p which map into some appropriate generating set for an algebra containhg the events ~(a,p) (a < ,O < wl), it can be seen that Q has a dense suborder of size NI. This if (Q, 5)is c.c.c., then applying MAH,to (QCW,5) gives the sequence xn : w, -+ R as CWTER5. MEASU.ALGEBRAS 61 specified in the hst conclusion. If (Q, 5)isn't C.C.C. then fk an uncountable sequence {pc : 5 < wi) of painvise incompatible conditions. Let FE be the domain of pc. Fix an uncountable set r C wi such that:

1. The family {Fe : E E r) forms a A-system with root F.

2. There exists an E > O such that for every in r and every a in FE,

~oE(4)> 1/2 + E-

3. For every Ej7 in I' and LY in F, the measure of the difference of pc(a)

and p,(a) iç less than e.

The last item follows from the fact that the family

generates a separable measure algebra for every y < wl and contains

whenever 7 is greater than max(F) and p is in 8.Now, for p~ and p, to be incompatible, it must be the case that, for some LY in FE\ F and 0 in F, \ F,

~(a,0) has measure at least E. Thus, for all E # 7 in r,

giving the second alternative of the theorem. CWTER5. MEASURE: ALGEBRAS

5.3 A Variation of Maharam's Theorem

Perhaps the most famous theorem in the area of measure algebras is the result of Maharam [42] which states that there are remarkably few of them.

Theorem 5.3.1. [42] There is a measure preseming isomorphism between any two homogeneous measure algebras of the same character.

If rc is a cardinal, then the two point measure on {O, 1) induces a measure on the product {O, 1)". The resulting measure dgebra is known as the

Haar algebra. It is easy to see that this measure algebra has character K and is homogeneous. Hence Maharam's theorem states that if ('R, p) is homo-

geneous measure algebra of character K, then there is a rneasure preserving

isomorphism between (R,p) and the Haar algebra of character K. F'rom this it is immediate that arbitrary measure dgebras are just copies of Haar algebras "glued together" with appropriate weights since any measure algebra contains a factor which is homogeneous in character. This section will focus on the following form of Maharam's theorem which is useful in the context of analyzing random graphs.

Theorem 5.3.2. Suppose (R,p) and (S,v) are measure algebras such that S has character everywhere greater than R and d is a complete subalgebra of R. If h : A + S is a measure preserving homomorphism then h extends to a measure preserving homomorphism h : R + S.

In [23] Frernlin presents a full proof of Maharam's theorem. Although this version of the theorem is not stated in his treatment, it is an immediate CHAPTER 5. MEASURE ALGEBRAS 63 consequence of the lemmas used there to prove the standard version of Ma- haram's theorem. TodorCeviC proved a special case of this theorem in [75] to solve problem DV on E'remb's list [24]. Theorem 5.3.2 was supplemented in his solution by the following fact.

Theorem 5.3.3. Suppose that h : d -+ B is a measure preserving homomorphism between two subalgebras of a measure algebra R. The map h fies An i3 iffh(a) 5 rms(a) for al1 a in dA.

Proof. Let h : d -t B fix AnB and a in d be given. Since h(a~c)= h(a) Ac for allcindril?,

h(a) 5 T4ns(h(a)) = GmB(a)-

For the other direction, note that us(a)= a if a is in A n B and since h is measure preserving and bounded by TA~B,h(a) mut equal a as weu. a

To demonstrate the usefulness of these two theorems, 1shall prove The- orem 4 of [75] using them.

Theorem 5.3.4. If I,fl are index sets and (R,p) is a measure algebra of character everywhere greater than #(I) #(Cl), U is an ultrafiter on 1, and fc (E E 0) is a sequence of elements of (R1/U,p) then there is a sequence cc (c E S1) of elements of R such that

1. p( f€)= p(cC) for al1 < in Cl and

2- whenever l? is a finite subset of S2 Proof. Let denote the constant sequences in (R'/u, p). Let A be a complete subalgebra of 7Z of character #(I) - #(O) containing fc for all E in O such that ~~(a)is in A whenever a is in A. By Theorem 5.3.2 there is a measure preserving homomorphism h : A -t R such that h(a) = a whenever ü is a constant map. By Theorem 5.3.3 h(a) 5 ra(a). Since %( f) is bounded by (the map which is constantly)

the elements cc = h(fE) (< E a)satisfy the theorem. Chapter 6

Applications of Random Forcing to Combinatorics

The program of analyzing the effects of forcing with a rneasure algebra breaks into (at least) two different areas: combinatorics and topology. Among the est results in this Line of research was Kruien's proof that Martin's Ax- iom for 0-linked posets is preserved by adding a random real. Kunen &O demonstrated that MAN,always fails after adding one random real by mod- ifying a construction of Roitman and showing that Souçlin's condition is not necessarily productive (see [49]). Laver later showed that forcing with a measure algebra does not produce Souslin trees (assuming MAH,in the gound model) . Barnett , in [2] and [3], analyzed the eEects of random reals on partition relations. In this chapter we will anaiyze the effect of random reals on a number of combinatorial statements. 6.1 Partit ion Calculus

Partition calculus is the study of more general forms of Ramsey's theorem. As its discussion is gxeatly facilitated by notation, it helps to first make a few definitions (see [20]for a more complete treatment).

Notation. Suppose that r,d are natural numbers and a,71, . .. , y, are order types, for i 5 r. The expression

abbreviates "for every coloring c : [ald+ (1, . . . , r), there is an i 5 r and a subset H of a of order type Yi such that c is constantly i on [Hld." If al1 of the 7;'s are the same then the notation becomes cr -+ (y):.

Ramsey's theorem in this language is simply w -t (w): for all d, r. There are many variations on this theme. The following will be of interest to us as well.

Notation. If a,p, "/o,yl are order types then n + (P, (yo : abbreviates

"for every coloring c : [al2-t {O, 1) either

1. there is a subset B & a! of order type ,B such that c is constantly O on [B]~or else

2. there is a pair of sets Co,Cl Ç a such that the order type of Ci is yi, Co < Cl, and c is constantly 1 on Cox Cl."

Sierpihki's famous partition [52]shows that c ft (wl)g. Hajnal showed in [30]that if the Continuum Hypothesis holds then even wl ft (wi,(w : 2))2 thus implying that the Dushnik-Miller relation wl + (wl, w f 1) is optimal (at least in ZFC). On the other hand Hajnal and Laver (see [40]) proved

IMAN, implies wi + (wl, (a: ~1))~for all a < wl.

Later, Barnett demonstrated in [2] that if MAR,holds then wl + (wl , (a :

~1))~stiU is true after forcing with a separable measure algebra. She notes the resdt of TodorEevïC (see [74] Proposition 6.4) which states that if there is a SierpiSski set then wi ft (wl, (w : w~))~.Bamett shows in [3], however, that if MAN, holds then wl + (wi,(a : n))2for ail a < wl and n < w after forcing with any measure algebra. In this section we will extend this result by replacing (a : n) by (a : a). Before proceeding, it wiu be usefd to develop some notation and make some definitions The motivation for the following definitions cornes from a class of ultrafilters which were introduced and developed by Hajnal and

Laver to show that MANlimplies wl -+ (wl, (a : w~))~for all a < wl-

Definition 6.1.1. A partition tree T (on a set of ordinals) is a collection of closed countable sets of ordinak such that the following conditions hold

1. T is a tree under the order of reverse inclusion.

2. If A and B are incompatible elements of T then either A < B (i.e. sup A < inf B) or B < A.

3. An element of T is a terminal node (Le. has no successors) ifE it is a singleton. 4. If A is an element of T which is not a terminal node, then A has infinitely many immediate successors, none of which contain max A.

5. If A is a nont erminal node of T then rnax A is the unique limit point of the collection of immediate successors of A (i.e. the minimums of the immediate successors of A forms an w-sequence converging to rnax A).

6- If B, C are immediate successors of a node A in T such that rnax B < min C then otp E 5 otp C.

Remark. This is a slightly different definition than the one usualiy associated with partition trees on linear orders.

Notation. The set of all a! such that {a) is a node of T will be denoted by TT.

Notice that all partition trees are well founded (Le. have no infinite branch) since the maximum huiction is a decreasing map into the ordinals dong any branch of a partition tree. The definition of a partition tree is made to facilitate the definition and discussion of classes of objects, usually filters, which are associated with them.

Definition 6.1.2. Suppose T is a partition tree. A filter 3 on S is a collection of partition subtrees of T such that

1. the empty tree is not in F,

2. if F0Ç .F is finite then nFOis in F, and 3. if S is in 3 and S' is a subtree of T which contains S, then Sr is also in F.

A filter F on T is tree-like if there is an association of a filter FA on the successors of A for each nonterminal A in T such that S is in 3 iff for every A in S, A has FA-many successors in T. A tree-like filter is an ultrafilter if the filters FAase all ultrafdters. A tree-like fdter 3is uniform if FAcontains the Fréchet filter for every A in T.

Notice that every filter F on a partition tree T induces a filter 7r3 on TT which is generated by (irS : S E 3).

Definition 6.1.3. The order type otp(T) of a pztition tree T is dehed by recursion on the rank of T. If T is just a singleton, then otp(T) = 1. If {&)=, is an increasing enurneration of the first level of T above the root of T then otp(T) = ~~=-,otp(T[A,])(where T[A]= {B E S : B Ç A)).

Tt is easily checked that, if T is a partition tree

whenever F is a uniform tree-like filter on T, thus justifying the definition of otp(T). It is well known and easily verified that for every indecomposable ordinal a,there is a partition tree T such that TT C cu and otp(T) = a. The following is essentidy proven in [40].

Lemma 6.1.4 (MAH,).If F is a unifom tree-Zike filter on a partition tree

T and {AE : E < wl} is a sequence of elements of TF then there zs an uncozmtable B (& wl such that nEEBAc has order type at least otp(T). CWTER6. COMBINATORTCS 70

The following standard absoluteness result will be very useful in proving the main theorem (see [6]). The proof is included for completeness.

Lemma 6.1.5. Suppose M 5 N are models of ZFC which contain the

same ordznals. If {Ac : 5 < wl) C M is a sequence of subsets of wl and a < wl is such that there are sets A, B C wl in N with A Ç neasAc and otp(A) = otp(B) = cr then M contains a pair of such sets which have the

same properties.

Proaf. Fix a bijection e : w -t a. Let (E, <) be the collection of a.lI pairs

(s,t) such that

1. s, t are injections from some set {O,. . . , n) into wl,

2. s(i) is in At(j)for every i,j 5 n, and

3. for every i < j 5 n, s(i) < s(j) iff t(i) < t(j) iff e(i) < e(j) and < orders E by coordinatewise extension. Clearly an infinite branch through (E, <) gives the desired object and (E,<) is well founded in M ifT it is in N. O

We are now ready to prove the main theorem of this section.

Theorem 6.1.6. Suppose that V rnodels MAH,and R is a measure algebra.

In V" the partition relation wi -+ (wl,(a: : a))2 holds for all a < wl.

Proof. Let a < wl be given and assume without loss of generality that a is indecomposable. Suppose that G is a name for a graph on wl. By Theorern 5-2.1 either G is forced to be countably chromatic or else there is a sequence FF (6 < wl) of disjoint n elements sets and a b > O such that for every E < 7 there exist i and j such that Fe@) and F,(j) are connected in G with probability at least 8. Let T be a partition tree such that TT C a and otp(T) = a. Let U be a uniform tree-like ultrafilter on T. Pick an uncountable set B C w, \ cr and a pair i, j 5 n such that whenever is in B, there are TU many J such that FE(i)and F, (j)are connected in G with probability at least 6. We will now confuse B with the set {F,(j): 7 E B} and T and U with the corresponding objects they induce on {FF(i): E a).

Fix a measure algebra (S,p) which contains R and has character everywhere greater then NI. Let T be a tree-like ultrafilter on A and set

for a E A and fi in B. Applying Theorem 5.3.4 repeatedly, find liftings fa : T -t S such that

2. the measure of &(A) equals

where A is a nonterminal node of T, and

3. for every finite l? Ç B and nonterminal A in T Define ~gby

[A E $1 = &(A) and A by

M E AI = &(root(~)).

It is easy to verify that U (s~: p E A) generates a tree-like Glter 0 on T. Let c E 'R+ force that A is uncountable. In VS fix a ccc. partial order (P,$) which forces MAN,.Applying Theorem 6.1.4 in the extension VS*P to {?r$ : ,û E A), it is possible to find sets Ào 2 TT and go A such that otp(Ào) = otp($) = o! and AO x BO 5 G (see [40]). Now applying Theorem 6.1.5, the desired homogeneous set can be pulled back to V". 0

It might initially seem that there is no reasonable positive partition relation which lies between wl -t (wl, (a : a))' and wl + (wl, (a: ~1))~-This, however, is not the case. Let (a) : ,û : y) denote the class of graphs of the form (A x B) U (B x C) U (Ax C) where A < B < C and otp(A) = a, otp(B) = ,!?, and otp(C) = y. It is clear that wl + (wl, (a : ~1))~implies that wl + (wl, (a : cr : which in turn implies that wl -t (wl,(a : a))2. This leads us to the following

Question 6.1.7. Does wl -+ (wl, (a : (Y : hold for w 5 a < wl in an extension of a model of MAH,by an arbitrary number of random reals?

Being even more ambitious, it is also reasonable to ask the following

Question 6.1.8. Does wl + (q,a)2 hold for w + 2 5 a < wl after forcing with a nontrivid measure algebra over a model of PFA? It should be remarked here that it is unknown whether MAN,implies even wl + (wl,w2 + 2)2 (see [22] and [24]).Therefore it is more natural to ask the Iater question under the assumption of PFA.

6.2 Uniformizing Ladder Systems

A ladder systern is a sequence of maps fA : CA -, w indexed by the limit ordinals X such that CA is a cofinal subsets of X of order type W. A ladder system CfA) is imiform if there is a map g : wl + w such that for each X fx(a) = g(a) for all but hitely many a in CA.Devlin and Shelah have shown in [14]that MAN,implies ail ladder systems are uniform. In this section we will discuss the influence of random reals on the exÏstence of nonuniform ladder systems.

Theorem 6.2.1. (MAN,) After forcing &th a separable rneasure algebra, al1 ladder systems are uniforml

Notation. If C C wl is a countable sequence of ordinals and n is an integer then let C/n denote the set C with its least n elements removed. Also, let C 1 n denote the least n elements of C.

Proof. Let { fx : A E A) be a given sequence of names cosresponding to a ladder system coloring where the domain of fA is CA. As in Theorem 5.2.1, define a partial order (Q, 5)as follows. Elements of Q are functions p such that

1. the domain of p is a finite subset of A x w, CHAPTER 6. COMBLNATORICS

2. the range of p is contained in R,

3. for all (or, m) and (p,n) in the domain of p, p(a, m) A p(,û, n) forces that f, and fB agree on (c~\ rn) n \ n), and

4. p(a,i) is disjoint from p(a, j) whenever (a,i), (a,j) are distinct el* ments of the domain of p.

The order < is defined by q 5 p iff

1- the domain of q contains the domain of p,

2. q(a,rn) < p(a, m) whenever (a,m) is in the domain of p, and

3. for al1 (a,m)in the domain of p, b(q(a,m))= b(p(a,m)).

Here if B is in R then &(B)is 2-k where k is minimal such that 2-C < p(B). Notice that for each p in Q there is a g, whkh is a name for a hinction fcom wl into w such that, for every (or, m) in the domain of p, p(a, m) forces that fa and gp agree on Ca \ m. If p is in Q, let ~(p)> O be the least rational (in some fixed enurneration) such that if (a,m) is in the domain of p then &(a, m))> b(p(a,m)) + ~(p).Note that if F is a subset of wl and p, q are conditions in Q such that

2. for every (a,m) in intersection of the domains of p and q, p(a, m) and q(a,rn) differ on a set of measure at most ~/3, 3. there is a A of measure at least 1 - ~/3which forces & n & C F for aU (a,m) in the domain of p and (p,n) in the domain of q, and

4. there is a B of measure at least 1-~/3which forces that g, r F = g, r F

then p and q are compatible.

Claim 6.2.2. (Q, 5)satisfies the countable chain condition.

Proof. Let CpE : E < wl) be a given sequence of conditions in Q. It suaices

to find a pair and 77 and a set F which satisfy conditions 1-4 above. Let I' C wl be an stationq subset such that the following conditions hold:

1- &(pE)= E&) for aU E, 7 in I' (condition 1).

2. {dom(pE) : < E l?) forms a A-system with root D.

3. If (cr,rn) is in the domain of pc and is not in D then a > E.

4. If E,q are in I' then p&m) and pq(a,m) differ on a set of measure less than ~/3for ail (a,rn) in D (condition 2).

5. If E < 7) are in I' then it is forced that C, is a subset of q whenever

(a,m) is in the domain of pc.

By 3 it is possible to find, for each E in I', a finite set FE contained in 5 and a condition Ac of measure at lest 1- ~/6which forces that fl is contained in FE whenever (a,m) is in the domain of pc. By pressing down on the FEys, it is possible to find an uncountable r' C I' such that Fc = F is the same for ail in î'. Notice that for every E # 7 in r', F and A = Ac A A, satisfy condition 3 for checking the compatibility of two conditions. By the separability of R, it is possible to hda B of measure at least 1 - 43and E # q in r' such that B forces gF F = g, r F. Thus E and 7 satis@ condition 4 and, since we have already ensured that they satisfy the other three, pc and p, are compatible. 0

Define V,, to be the collection of all p in & such that for all a in A for which (or, m) is in the domain of p for some m,

where ml, . . . ,rnk list the integers m for which (ai, m) is in the domain of p.

It is routine to ver% that 'D,,, is dense for all CI,E.

If 3L is a ater intersecting all of the Va,E'sthen dehe P : A x w -t R by

P(a,m) = ~{p(a,m) : p E X and (a,m) E dom@)).

In V", P defines a function P fiom A to w by ~(a)= m where m is the unique integer such that P(a,m) is in the generic filter ({P(a,m)}, is a maximal antichain if 31 meets Da,for all E). Define jr(E) = f,(~) if E is in c,/P(~)for some or and arbitrarily otherwise. By condition 3 in the definition of Q, g is well defined and dearly g uniformizes the ladder system. Cl

Fact 6.2.3. If R is a homogeneozcs nonseparable measure algebra then there is a nonuniform ladder system in V". Remark: 6.2.4. This can be established by some slick arguments using the inequality 2N~< 2*' and a theorem of Devlin in 1141 but 1 shaii give a direct proof.

Proof. Let + : wl + {O, 1) be a random subset of wl and let {CA)be any sequence of ladders in the ground model. Dehe fx on CAto be the constant

value +(A). Suppose that g : wl + {O, 1) is a name for a potential uniformizing function. By closing off under some appropriate function, it is possible to fbd a limit ordinal 6 < wi such that T(6) is undetermined in V[g r 61. Now it is easy to choose a condition which forces +(6) to have a different value than the value g eventudy attains on Cs (if there is such a value). Cl

6.3 Wage's Lemma

Consider the following combinatorid statement for a regular cardinal 0:

W(0) If A is a family of countable subsets of 0 such that every pair has a finite intersection, then there is a cofinal subset of 19 which has finite intersection with every element of A.

This statement is known as Wage's Lemma and was considered by Wage in [87] through the course of working on topological problems related to S and L spaces. It also has uses in combinatorics - the interested reader is referred to [87]. In this section we shall interest ourselves in proving that W(8) is a comequence of MAe which caxi hold after forcing with an arbitrary measure algebra. Theorem 6.3.1 (MAo). W(8)holds after forcing vith any rneasure algebra.

Proof. Let (A( : < 6) be a sequence of R-names for almost disjoint countable subsets of 8. For each a < 8 let Rabe the separable complete subalgebra of R generated by {I[c E A,I : 6 < 0) (note that [< E A,B is O for all but countabiy many E). Let (Q, 5)be the forcing notion of au pairs (X, P) which satisfy the following properties:

1. X is a finite subset of 13.

2. P is a finite partial map Born 8 into R+.

3. If (p,M) is in the dornain of P then P(B,M) is in Rg.

4. P(P,M) forces that X n A, has size at most M.

The ordering on elements of Q is defined by q 5 r if the foUowing hold:

2. The domain of P, contains the domain of PT.

3. For every (B,M) in the domain of PT,b(PT(Bj M)) = 6(Pq(P,M)).

As in the section on random ladder systems, for each condition (X, P), there is a rational E = E(X,P) > O such that p(P(P,M)) > 6(P(P,M)) + E.

Claim 6.3.2. (Q, 5)satisfies the countable chain condition. Proof. Let {(Xe,PF) : E < wl) = {qc : E < wi) be a sequence of conditions

in Q. Select an uncountable l? C wl and an E > O such that the following conditions hold:

2. The families {XE: 6 < wi) and {dom(PF): E < wi) form A-systems with roots X and D respectively. Moreover Xe n ,$ = X and dom(PE)rl Cxw=D.

3- If (p,M) is in D and (,q < WI then PF(P)M) and PT(@,M) diEer by a set of measure at most ~/2.

4. If < 7 are in l? then XE Ç 17 and it is forces that is contained in q for aii (B,M) in dom(PE).

Let m denote the cardinality of Xc \ X and 1 be the fkst w elements of r. For 7 > sup I and i 5 rn, define fi : I I' R so that f;(E) is the event that the ith element of XE \ X is in A, for some (a,M) in the domain of P,.

Notice that if î) # C are in I' \ 1 then p(f; (E) - fj (E)) Mnishes for a.ll i as < + sup 1. In particular, this means that for some 7 in I'\ 1 and some J in I p(fi(~))< ~/m for all i. It is now easy to verify that (XE,PF) and (X,,P,) are compatible. O

The proof is now finished with an application of MAs to (Q, 5)and some suitably chosen dense sets (see the proof of Theorem 6.2.1). 0 CHAPTER 6. COMBINATORICS

6.4 Fragments of Martin's Axiom

The very first results concerning random reals and Martin's Axiom were due to Kunen and concern the effect random reals have on the standard kagments of Martin's Axiom-

Theorem 6.4.1. (see [dg]) The addition of one random real preserves Mar- tin's Axiorn for a-linked partial orders.

By making a variation on Roitman's result for Cohen forcing Kunen showed that MAN,always fails after adding a random real:

Theorem 6.4.2. (see [49]) After the addition of a single random real, theie zs compactum K such that K satisfies Souslin's condition but K2 doesn't.

It is still unclear whether this result is optimal.

Question 6.4.3. (MAN,) After adding one random real, does MAN, hold for all partial orders whose countable chain condition is productive?

The situation after adding many random reals (i.e. forcing with a homogeneous nonsepasable measure algebra) is much different. In this model, there is dways a SierpiEski set and a covering of IR by NI many nowhere dense sets (hence MAN,fails even for countable partial orders). It can also be shown that there is a disparity between most of the local chain conditions (e-g. the example fiom Section 3.2 shows that there a compactum which has Knaster's condition K2but not Knaster's condition K3). Kunen7s example mentioned above demonstrates that after even one random real there is an example of a compactum which satisfies Souslin's condition but not Knaster's condition.

What was less clear until recently was whether there was an example which distinguishes between the property of having the C.C.C. productively

(i.e. the product with any other C.C.C. partial order is c.c.c.) and Knaster's condition. We will see in Section 7.1 that there is always such an example in the presence of a Sierp&ki set. A Sierpiliski set with the density topology gives an example of a cometrizable L space (see Section 7.1). The naturd partial order for forcing a discrete set through such spaces is dways C.C.C. and, since being cometrizable is absolute, productively C.C.C.(see the Claim on page 62 of [74]). The partial order, however, will not satisfy Knaster's condition. In [66] TodorCeviC asks whether the nonexistence of a nonseparable perfectly normal compactum (i-e. a compact L space) is equivalent to a "standard fkagment" of MAu,. In [75], he has shown that if MAN, holds then forcing with any measure algebra does not introduce a compact L space. At this time, 1 am unaware of any fragment of MAN, which is considered standard that is not known to fail after forcing with a nonseparable measure algebra (except perhaps SMN,,which was designed for the purpose of killing compact L spaces). Chapter 7

Applications of Random Forcing to Topology

Two basic properties one encounters in general topology are separability and the Lindelof property. Here a space is Lindelof if every open cover has a countable subcover. Thus it is a dual notion to separability. These two properties are hereditary in the class of all metric spaces as they are both equivalent to having a countable base in th% context. A natural question to ask is whether being hereditary separable is equivalent to being hereditarily Lindelof in the broader class of all regular spaces '. Notice that a naturd obstruction to being either hereditarily Lindelof or hereditarily separable is for a space to contain an uncountable discrete subspace. The above question breaks naturally into two halves.

(S) 1s every hereditarily separable space hereditarily Lindelof? 'AU spaces considered in this thesis are regular. CHAPTER 7. TOPOLOGY

(L) Is every hereditarily Lindelof space hereditarily separable?

Both of these questions are essentiaily Ramsey theoretic in nature (see [SOI). A counterexample to (S) has become known as an S space and a counterexample to (L) has become known as an L space. Given any non Lindelof space

X it is easy to find a sequence of points (x, : a < wl)in X such that every final segment of this sequence is relatively closed. Such a sequence is said to be right separated and, as one might expect, nearly every study of an S space immediately passes to such a sequence. Similarly, nonseparable spaces contain such a sequence in which every initial segment is relatively closed (these sequences are said to be left separated). Eventually a positive answer to (S) was shown by TodorCeviC. [70] to be consistent (and a consequence of PFA). Whether (L) can have a positive answer is still unknown. It is known, however, the a positive aoswer to (L) does not follows fkom (S) and that MANl does not have enough strength to provide a positive answer to either question (see [74]). Much earlier, a number of counterexamples to both (S) and (L) appeared, following fkom a variety of axioms such as 0,4, and CH. Typicdy these spaces had additional properties such as compactness or cometrizability (see e.g. [47], [16], and [36]). Even though obtaining consistent positive answers to (S) and (L) turns out to be a somewhat difficult task for which MAN,is not sufficient, MAu1typicdy gives a positive answer to (S) and (L) in more restrictive classes of topological spaces. For example, in the same article in which Szymariski constructed his extremally disconnected S space fiom 4, CRAPTER 7. TOPOLOGY 84

he &O demonstrated that Wage's lemma (a consequence of MAN,)implies that there are no extremaIly disconnected S spaces. It was similady shown by Gruenhage [28] that MAN, implies a positive answer to (S) and (L) in the class of cometrizable spaces. Szentmikl6ssy [60] has shown that MAR, implies a positive answer to (S) and (L) in the class of ail compact spaces. On another front, a program has been initiated to forward our understanding of which consequences of MANl are presemed by forcing with a measure algebra. This began with Laver's result in [41] that there are no Souslin lines in an extension of a mode1 of MAN, by random reals. In [75] TodorCevié extended this t heorem by replacuig "Souslin linen with "compact L space." The reasons for analyzing the effects of random reals on consequences of Martin's Axiom began more or less as a curiosity. It was realized, however, that this scenario has the potential for demonstrating that certain consequences of MAN,(particularly those concerning (S) and (L)) are consistent with the inequality 2N0< 2". The assumption 2N0< 2'1 can have a strong impact on the normality of certain topological spaces and it is therefore of great interest to general topologists to know when tbis inequality can hold in conjunction with some particular instance of (S) or (L). In this chapter we wiU analyze the influence of random reals on (S) and (L) in the classes of cometrizable spaces, compact spaces, and extremally disconnected spaces. CRAPTER 7, TOPOLOGY

7.1 Cornetrizable Spaces

A topological space X is cornetrizable if there is a weaker metric topology on X such that every point of X has a neighborhood base of sets which are closed in the metric topology on X. In this section we will consider the idLuence of random reals on (S) and (L) in the class of cornetrizable spaces. It turns out that whether these objects are introduced by forcing with a measure algebra depends on the character of the algebra.

Theorem 7.1.1. After adding one random real to a mode1 of MAH,(S) and

(1;) have a positive answer in the class of cornetrizable spaces.

Proof. 1 will only present the proof for (S) as the proof for (L) is symmetric.

Suppose that x = {IÉ : E < wl) is an enumeration elements of a metric space (x7d) in Va and (4: [ < wl) is a sequence of names for closed sets such that ËFn{kq : 9 > <)is empty for all E < wl. Let i denote the topology on x defined by declaring the sets ÈF to be clopen. Dehe 6 : [wl12-t R, by ~(a,B) is the event "xa is in Ëp" where cr < p < wl. By Theorem 5.2.1, either G is forced to be countably chïomatic (and hence (2,i) is forced to be 0-discrete) or else there is an uncountable sequence of disjoint n element sets FE (6 < wl) and a 6 > O such that for alle # 0

We will show that the second alternative can not hold. Suppose that such a sequence FE (5 < wl) and a 6 > O are given. For each 5, pi& a rational CHAPTER 7. TOPOLOGY

EE > O such that

Ac = [min min d(*p, E,) > ~€1 aEF.PEFC+l

has measure greater than 1 - 6/2. Fix an uncountable r Ç wl and E > O

such that EE = E for all E in î. By the separability of 72, it is possible to find [

has measure greater than 1 - 6/2. It follows that A, A BE,has measure greater than 1 - 6 and forces that d(x,, Ëp)> O whenever a is in and ,O is in F, and hence is disjoint from

In [89] White showed that a Sierpinski set with the density topology is an L space (see also [62]). If the SierpSski set is viewed instead as a subspace of the Stone space of the measure algebra then it is an example of an extremally disco~ectedL space, a fact that will contrast the result in Section 7.2. A consequence of the Luzin-Menchoff theorem (see [62]) is that the density topology on the reals has a base of sets which are closed in the real topology. Thus a SierpZski set with the density topology is a cornetrizable L space.

Taking this a step further, suppose that {x,),,, Ç [O, 11 is an enumeration of a SierpZski set which moreover has the property that it is hereditarily Lindelof in all hite powers in the density topology (an wl sequence of random reals is such a set). For each (Y < w1 pick a compact set E, of positive CRAPTER 7. TOPOLOGY 87

measure which misses xT for all y < a: but which contauis x, as a point of

density 1. Now the sets U, = {y < wl : x, E ET)are closed in the Vietoris topology on the closed subsets of [O, 11 and, by duality, generate an S space

topology on {E, : o: < wl). Thus there are always cometrizable S spaces after forcing with a nonseparable measure algebra.

In [74], TodorEevZ has shown that b = Hl implies that there is a locally compact cometrizable S space topology on any wl sequence of reals.

Question 7.1.2. 1s it possible to construct a locally compact locally countable cometrizable S space fkom wl random reals?

A closer exmation of Todorëevié's construction shows that it is very

closely connected to the negative partition relation wi f, (wl,(w : 2))'.

Since, by a result of Barnett [3], wl + (wl,(w : 2))2 can hold after adding

wl random reals, if this question has a positive answer, then some new ideas are required.

7.2 Extremally Disconnected Spaces

A topological space X is extremally dz'sconnected if the closure of every open set is open. Extremally discomected S spaces were constructed by Ginsburg [W],Szy-ma1isk.i [61], and Wage [88] using a variety of set theoretic assump tions. In the last of the three papers written on extremally disconnected S spaces, Szyma6sk.i shows that Wage's Lemma, a knom consequence of MAN, (see [87]),implies that (S) has a positive answer in the class of all subspaces CWTER 7. TOPOLOGY 88 of extrernally disconnect spaces. In Section 6.3 we extended Wage's result, showing that if MAN,holds then Wage's Lemma remains true after forcing with any measure algebra. Combined with the results above, this estab- lishes the consistency of "(S) has a positive answer for all subspaces of extremally disconnected spaces" with statements such as "there is a Sierpiriski set" (which yields an extremdy disconnected L space) and "there is an Ostaszewski spacey' (see Section 7.3). The latter is perhaps surprising since historicdy extrema.lly disconnect S spaces and Ostaszewski spaces have been constructed fkom similar axioms. 1 wiU close this section with the proof of Szymahski's theorem.

Theorem 7.2.1. [61] (W(N 1)) Exhernally disconnected spaces can not contain S spaces.

Proof. Suppose that X is extremally disconnected and that {x, : a < wl} is a right separated subspace of X. Recursively choose countable sets A, (c < w,) such that

1. A, C q is almost disjoint koom AE for every E < 7,

2. {xc : E E 4)does not accumulate to any point in {XE : < wwl)except for possibly x,, and

3. if possible, (XE : E E 4) accumulates to x,.

Now suppose that F C X is almost disjoint kom every At for 6 < wi. Suppose for contradiction that F does accumulate to some point xc and let be minimal with this property. Let W be a neighborhood of xt in X whose CHAPTER 7. TOPOLOGY 89 closure misses the closure of {x, : 1) > c). Since Ac n F n W is finite and Ac and W n F are countable, it is possible to pick disjoint open sets U and V in X such that U contains all but finitely many elements of AE and V contains all but fhitely many elements of F nW. Since X is extremaXy disconnected, the closures of U and V are disjoint. But this is a contradiction since XE is an accumulation point of both Ac and F n W. O

7.3 Compact S Spaces

One of the primary reasons for studying (S) and (L) after forcing with a measure algebra is that thîs may yield a solution to an old problem of Katëtov. In [35] Katétov proved that if X is a compact space and X3 is hereditarily normal then X must be metrizable. He then asked whether dimension 3 could be lowered to dimension 2. Gruenhage and Nyikos have shown in [29] that under CH and also under MAN, this question has a negative answer. Moreover they show that if X is a counterexample, one of the following must hold:

1. There is a Q set.

2. X is a compact L space.

3. X2 is a compact S space.

4. X2contains both S and L spaces. CIXAPTER 7. TOPOLOGY 90

After forcing with a measure algebra of character at least there are no Q sets. Also, TodorEevik has shown the following.

Theorem 7.3.1. [75] (MA*,) After forcing vith any measure algebra every compact space containing an L space also contazns an uncountable free sequence.

Thus any counterexample to Katëtov's problem in this model would have to be a perfectly normal compactum X whose square is an S space. For some time it was uncleu whether or not TodorEevE7stheorem codd be dualized by replacing "S space" by "L space." The following theorem gives a negative answer to this question.

Theorem 7.3.2. In any forcing extension by a homogeneous nonseparable measure algebra there is a perfectly normal countably compact non compact topology on wl. In particular, such forcing extensions always contain compact S spaces.

Remark 7.3.3. Such a space is often known as an Ostaszewski space (see [47] for Ostaszewki's construction of such a space under 0)-Such a space must be an S space. Eisworth and Roitman [17] have shown that such spaces can not be constructed from CH alone. This is the only example of an

"anti-Ramsey" object on wl that 1 am aware of which is produced by an wl sequence of random reds but not by CH.

Proof. By modieng our ground model if necessary, we may assume that

Va= V[Fa: cr < wl]for some sequence of random reals Ta (a< wi).It will CHAPTER 7. TOPOLOGY 91 be convenient to view T, as a random reaI in wW where w is given the atomic measure determined by p({n))= 2-"-'. In V fix, for each limit ordinal 6, a strictly increasing sequence &, cohal in 6. Define a sequence of topologies & on the limit ordinals a by recursion so that .i, is locdy compact, non compact topology in V[fc : E < a] and .ip r a is i, for a < B. i,is the discrete topology. Suppose now that fa has been defined (limit stages are trivial). Dehe a topology i,+,on or + w as follows. In V[+ : E < a], let

{Üa(k) : k < w) be a partition of (a,fa) into disjoint compact open pieces.

Neighborhoods of ct + n in fa+" a.re of the form {a + n) u U v where v is a cofinite subset of

{&(k) : fff(k)= n).

The space we are interested in is, of course, (wl,i,, ). The topology is clearly Iocally compact and locally countable. It should also be clear that the genericity of the Ta's ensure that, for a hed ac < wl, the closure of {a + n : n < w) is the set of all p 2 a. We will now show that (wi,i,,) has the property that the closure of any set is either compact or cocountable.

Suppose that Ë is a name for an i.tesubset of wl and assume without loss of generality that Ë is forced to be either countable or uncountable. Let a be a lMit ordinal such that & n a is infinite and is in V[Tc : E < O].If E is forced to be uncountable, then also arrange that E n a is forced to be cofhal in a. Now we will work in V[fc : 6 < a]. If È n ru ha. compact closure in (a,7,) then we are done (note that this is impossible if En ru is cofinal in a). If È n a does not have compact closure in .i, then there are hfhitely many CHAPTER 7. TOPOLOGY 92

k such that Ù,(k) fi E is nonempty. Moreover, an easy genericity argument shows that, for each n there are in6niteIy many k such that &(k) n E is non empty and &(k) = n. It follows that È n or accumulates to each of the

a! + n's which are in turn dense the set of aU ,û 2 a. Cl

7.4 Small Compactifications of L Spaces

By a result of Fremlin [22, 44A] (see ais0 Section 6 of [66]), MANlimplies that every compact space containing an L space must rnap onto [O, Ilw1.In light of TodorEevit's Theorem 7.3.1, it is natural to ask whether this result dso holds after forcing with a measure algebra. The following result indicates that this is not so.

Theorem 7.4.1. If there is a SierpZski set then there is a compact space K which contains an L space and which is also linearly fibered (and hence does not rnap onto [O, 1IW1).

Proof. Let {x, : or < wl) be a Sierpiriski subset of 2" and let e, : a -i w (a < wl) be a coherent sequence of injections. Suppose further that for ail

a < p A(x,, xp) 5 e&) (the sequence e, can always be modified to have this property - see Chapter 4 and in particular page 96 of [y]). Defhe

Then Ep is a compact set of positive rneasure all of whose elements are of Lebesgue density 1 and which contains xp. Hence if a topology on 2W is CHAPTER 7. TOPOLOGY 93 generated by the clopen sets and the Ep7s,X = {xa : a < wl) is an L space when given the subspace topology. Let L3 be the BooIean algebra generated by the clopen subsets of 2" and the EB's.

Claim 7.4.2. The map from the Stone space of B to 2" given by restricting ultrafilters to the algebra of clupen sets has linear fibers.

Proof. Suppose that x is in 2". Let r, be the collection of all fl such that x E Ep- We must show that if a < ,û are in J?, then, for some neighborhood U of x, Eg n U & E, n U. To this end, it suflices to show that Ep \ E, is closed. Let F cu be a finite set such that e, and ea agree on a \ F. Then

is clearly closed. Appendix A

Glossary

A < B If A and B are subsets of some linear order (L,<) then A < B abbreviates a < b whenever a is in A and b is in B. b The cardinality of the smallest unbounded subset of (RIN, <*)'). c The cardinality of RI cf's The Mage of S under the map c.

A (x,y) If x and y axe two distinct sequences then A (2,y) is the least n such

that x(n) # y(n).

N The set {O, 1,2,. . . , ) of natural numbers. Identical to W.

[y" The collection of all infinte subsets of N.

P The collection of all strictly increasing functions fkom N to N.

R, The homogeneous measure algebra of character W.

94 APPENDIX A. GLOSSARY

#(S) The cardinaIity of the set S.

Itl The length of a finite sequence t.

tni The concatenation of the sequence t with the entry i.

[]The event "4 is true" in the context of forcing with a measure algebra.

x <' y If x and y are in N~ then x <* y means that x(n) < y(n) for ail but

finitely many n.

x <" y If x and y are in IVN then x _ m.

[XIk The collection of aJ k element subsets of a set X.

almost disjoint Two sets A and B are almost disjoint if their intersection is finite.

analytic A set of reals is analytic if it is the continuous image of a Polish space.

Boolean algebra A Boolean algebra is a ring with 1(B, +, -) in which b2 = b for every b in B. Usually one defines aAb = a-b and avb = a+ab+b.

Also, a 5 b if a /\ c = O whenever b A c = O and the complement a" is

the unique element of 13 such that a V a" = 1 and a A a" = 0. centered A collection of sets is centered if it has the finite intersection property. A Boolean algebra is O-centered if it can be decomposed

int O count ably many centered families. APPENDK A- GLOSSARY 96 character (of a measure algebra) The character of a measure algebra is cardinaJi@ of the smdest set which completely generates it (via com- plementation and arbitrary supremurns and infimirmç). clique A clique is a synonym for a complete graph - the edge set consists of all unordered pairs of vertices. cometrizable A topological space X is cometrizable if there is a weaker metric topology on X such that every point in X has a neighborhood base of sets which are closed in the metric topology. complete (Boolean algebra) A Boolean algebra is complete if every fam- îly has a supremum and an infimm in the algebra.

Continuum Hypothesis (CH) The Continuum Hypothesis is the state-

ment that R can be well ordered in type wl. cover (of a measure O set) A cover of a measure O set G C 2" is a se-

quence Fn (n < w) such that Fn C 2", every element of G is in infinitely

many of the clopen sets [F,],and #(Fn) 2-n < W. countable chain condition (c.c.c. ) A partial order satisfies the countable chain condition if every uncountable subset collection contains two elements with a common lower bound. countably chromatic A graph G C [XI2 is countably chromatic if there

is a map f : X -+ w such that {x,y) are not an edge of G whenever x # y aze in X and f (x)= f (y). APPENDDCA. GLOSSARY

A-system A collection of hite sets F forms a A-system if there is a finite set R (called the root) such that A n B = R whenever A and B are distinct elements of F and a.U of the members of .F have the same size.

dense (in a hear order) A linear order (L, <) is dense if for every a < b in L there is a c in L such that a < c < b.

dense (in a partial order) A subset 23 of a partial order (P,5) is dense if for every p in P,there is a d in 2> such that

dense (in a topological space) A subset D C X of a topological space if dense if D has nonempty intersection with every nonempty open subset of X.

density topology The density topology is the topology on the reals in which the neighborhood at a point x is the collection of all Borel sets of positive measure which contain x as a point of Lebesgue density 1. For more information about this topology, see [62]. discrete A subset E of a topoIogica1 space is discrete if for every point x in E, there is a neighborhood of x which intersects E only at {x). event An event is any element of a measure algebra. The event ES], where S is a sentence concerning objects in a forcing extension by a measure algebra, is the unique element of the measure algebra which forces S is true and for which the complement of ([Sn forces that S is false. APPENDLX A, GLOSSARY 98 factor A factor (Ra,pu) of a measure algebra (R,p) the restriction of 'R to the (positive) element a. Here pU(c)is dehed to be p(a - c)/p(a). filter A collection 3 of subsets of a set S forms a fdter if A fî B is in 3 whenever A and B are in 3 and C C S is F whenever C contains a member of 3. Filters on a Boolean algebra are defined in an analogous

waF A subset Ç of a partial order (P,5) is a filter if every pair of elements in Ç has a lower bound in Ç and p E P is in G whenever p is

greater than some q in Ç. free sequence A sequence of points (x, : o: < b} in a topological space is

a kee sequence if for every y < b the closures of {x, : cr < y} and {xD : fl 1 7)are disjoint. Such a sequence is necessarily discrete.

Haar algebra The Haar algebra of character K is the measure algebra corresponding to the product measure space (2",p) where each factor 2 = {O, 1) is given the uniform probability measure.

Helly9scondition A topological space (or Boolean algebra) satisfies Helly's condition if it has a base B such that every linked subcollection of B is centered. homogeneous (for a partition K) If K C [SI2is a partition then H is K

homogeneous if [Hl2 is containecl in K. Similarly, if c : [SI2+ Z is a coloring, then H is homogeneous for c if c is constant on [Hl2. H is i homogeneous if c is constantly i on [Hl2. APPENDLX A. GLOSSARY 99

homogeneous (measure algebra) A measure algebra (R,p) is homogeneous if every factor of R has the same character-

ided A collection Z of subsets of N is an ideal i8F it is closed under unions and subsets.

indecomposable An ordinal a is indecomposable if a can not be expressed as the sum of two smaller ordiaals.

irreducible A coloring c : [xI2+ Z is irreducible if d'[YI2 = Z for every Y E X with #(Y) = #(X).

K,, This is the assertion that every uncountably collection of elements of a Boolean dgebra must contain an uncountable n-linked famiI.y.

Katetov's problem Katetov asked if X is metrizable whenever X is a compact space such that every subset of X2 is normal.

Knaster's condition Kn A TopoIogical space satisfies Knaster's condition Knif whenever F is an uncountable collection of open sets .F contains an uncountable n-linked subcollection.

L space An L space is a regular topological space which is hereditarily Lin- delof but not hereditarily separable. Every L space contains a left separated sequence of length wl . ladder system A ladder system (on wl) is a sequence fA of functions indexed by the limit ordinals such that the domain of fx is a cofinal APPENDX A. GLOSSARY 100

sequence in X of order type w and the range of fA is contained in the natural numbers. left separated A sequence {x, : cr < wl) in a topologicd space is left

separated if for every 7 < wi, the closure of {xa : a < 7)is disjoint

from {xp : ,û 2 7).

Lindelof A topological space is Lindelof if every open cover has a countable subcover. linear order A linear order is a pair (L, 5) such that L is a set and 5 is a relation which is reflexive, mtisymmetric, transitive, and satisfies dichotomy (for every a, b in L either a 5 b or b 5 a). linear compactum A linear compactum L is a compact topological space such that for some order $ on L the open intenmls of (L, 5)form a base for the topology on L. linearly fibered A compact topological space K is linearly fibered if there

is a continuous map 4 : K -t [O, 11 such that the preimage of every point in [O, 11 is a linear compactum.

Iinked (n-Iinked) A collection of sets F is linked (respectively n-linked) if every pair (respectively every n element set) of elements has a nonempty intersection.

Martin's Axiom (MA) This is a forcing axiom for the class of ail C.C.C.

partial orders. It asserts that for every C.C.C. partial order and every APPENDXX A. GLOSSARY 101

sequence of fewer than c many dense sets, there is a filter meeting each of these dense sets. MAo, for a cardinal 8 is the above assertion where "fewer than c dense sets" is replaced by "at most 0 dense sets." Further reading on Martin's axîom can be found in [7, Ch. 31, [79], [37], and

V4I-

measure algebra A measure dgebra is a pair (R, p) where R is a complete

Boolean algebra and p : 72. + [O, 11 is a strictly positive map such that

~(1)= 1 and dVr0ai) = CEop(%) whenever ai A Uj = O for all i #j.

metrizably fibered A compact topological space K is metrizably fibered

if there is a continuous map 4 : K + [O, 11 such that the preimage of every point in [O, 11 is metrizable. r nice A cover F of a measure O set G E 2W is r nice iflim+, #(&)iz-'/2' = O. nowhere dense A set is nowhere dense if it is not dense in any nonempty open set.

Open Coloring Axiom (OCA) This is the assertion that whenever G [XI2 is an open graph on a separable metnc space - G is an open subset of [XI2 - then either G is countably chromatic or else there G contains an uncountable clique. orthogonal (modulo Z) Two subsets a and b of N are orthogonal modulo an ideal Z if an b is in 2. Two families A and B are orthogonal modulo Z if every element of A is orthogonal to every element of B.

T-base A T-base of a topological space X is a family B of nonempty open sets such that every nonempty open subset of X contains some element of B.

T-character The n-chaxacter of a point is the size of the smallest T-base at that point.

P-ideal An ideal Z is a P-ideal if every countable sequence of elements of Z is almost contained in a single element of Z.

Proper Forcing Axiom (PFA) This is a forcing axiom for the class of all proper partial orders. A full discussion of proper partial orders and PFA is beyond the scope of this thesis. For our purposes it will suffice to know that the proper partial orders encompass both the C.C.C.and the a-closed orders and that this class is closed under finite iterations. Interested readers are referred to [7, Ch. 31, [79, Ch. 121, and [74] for a concise sarnpling of some of the state of the art consequences of PFA in the context of Ramsey theory as well as an introduction to this axiom. [56] provides probably the most complete treatment of PFA and proper partial orders.

Polish space A topological space X is a Polish space if the topology on X is compatible with a complete separable metric.

Q set A set of reak X such that every subset Y of X is a Gs set in the subspace topology on X. random real A generic mter (sometimes construed as a real number) for the Lebesgue measure algebra. right separated A sequence {x, : cr < wl) in a topological space is right

separated if for every 7 < wl, {x, : a < y) is disjoint fiom the closure

of {q : P 2 7)-

S space Any space which is hereditarily separable but not hereditdy Lin-

delof. Every S space contains a right separated sequence of length wl. separable (for measure algebras) A measure algebra is separable if it has countable character. The Lebesgue measure algebra is the unique atomless separable measure algebra. separable (for topological spaces) A topological space is separable if it has a countable dense subset.

Shanin's condit ion A topological space satisfies Shanin's condition if every uncountable collection of open sets contains an uncountable centered collection.

Souslin's condition A topoiogical space satisfies Souslin's condition if every collection of pairwise disjoint open sets is countable. This condition

is &O known as the countable chah condition.

Souslin's problem 1s (B, <) the only dense, complete hear order with no hrst and last elements which satisfies SousIin's condition? APPENDLX A. GLOSSARY 104

f splitting A set X C fl is f splitting iff T does not contain an f thin subset of the same cardinaliw.

splitting node If X C wu and t is a finite sequence of natural numbers then t is a splitting node of X if there are infinitely many i such that t^iis an initial segment of some x in X.

subtree A subtree of a tree T is a subset of T which is &O downwards closed-

f thin A set X C ww is f thin if for aU but fmïtely many n in w and every t in wn there are at most f (n) many i for which ti is an initial segment of some x in X. tree A tree (T, <) is a partially ordered set in which {s E T : s < t)is weii ordered by < for every t in T.

Tychonoff cube The Tychonoff cubes are the spaces of the form [O, 11' where 1 is any index set.

Wage's Lemma W(0) W(0) is the statement that when ever A is a collection of almost disjoint countable subsets of 8 there is a cofinal subset of 0 which is almost disjoint kom every member of A. welI founded A pastial order (P,5) is well founded it does not contain any irifinite descending chah. Bibliography

[II U. Abraham, M. Rubin, S. Shelah. On The Consistency of Some Parti- tion Theorzms for Continuous Colorings, and the Structure of N -dense Real Order Types. Ann. Pure Appl. Logic 29 2 (1985) 123-206.

[2] J. H. Barnett. Eflect of a Random Real on K + (K, (a: ~1))~.Periodica Mathematica Hungarica 30 1 (1995) 27-36.

[3] J. H. Barnett. Random Reals and the Relation wl + (wl,(a : n)l2. Periodica Mathematica Hungarica 30 3 (1995) 171-176.

[4] T. Bartoszy&ki, H, Judah. Set Theory: On The Structure of the Real

[5] J. Baumgartner. Al1 NI-Dense Sets of Reals Can Be Isornorphic. hd. Math. 79 (1973) 101-106.

[6] J. Baumgartner, A. Hajnal. A proof (involving Martin's An'om) of a partition relation. bd.Math. 78 (1973) 193-203.

[7] M. Bekkali. Topics in Set Theory. Springer-Verlag (1991). [8] M. Bell. Compact C.C.C. nonseparable spaces of small weight. Topology Proceedings 5 (1980) 11-25.

[9] M. Bell. A Compact c. c. c. nonseparable space from a Hausdor-gap and

Martin's axiom. Comment. Math. Univ. Carolinae 37, 3 (1996) 589-594.

[IO] B. Bollob&. Cornbinatorics. Cambridge University Press (1986).

[Il] G. Cantor. Beitrage Zur Begriindung Der fiansfiniten Mengenihre. Math. Ann. 46 (1895) 481-512.

[12] G. Cantor. Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications (1952).

[13] K. Devlin. Constructibîlity. Springer-Verlag (1984).

[14] K. Devlin, S. Shelah. A Weak Version of O Which Follows Rom 2'0 < 2'1. Israel Journal of Math. 29, 2-3 (1978) 239-247.

[15] E. K. van Douwen. The Integers And Topology, in Handbook of Set-Theoretic Topology (K. Kunen, J . Vaughan eds .) . North-Holland (1984) 111-167.

[16] E. K. van Douwen, K. Kunen. L-spaces and S-spaces in P(w). Topology Appl. 14 (1982), 143-149.

[17] T. Eisworth, J. Roitman. CH With No Ostasiewski Spaces. Transi. Amer. Math. Soc. 351, 7 (1999) 2675-2693.

[18] R. Engelking. General Topology. PWN, Warszawa (l977). [19] P. Erdos. Some Remarks On The Theory Of Graphs. Bull. Amer. Math. soc. 53 (1947) 292-294.

[20] P. Erdos, A. Hajnal, A. Maté, R. Rado. Combinatorial Set Theory: Partition Relations For Cardinals. North-Holland (1984).

[21] 1- Farah. Analytic Quotients: Theory of Liftings for Quotients Over Analytic Ideals on the Integers. Mernoirs of the AMS, to appear (2000).

[22] D. H. Fremlin. Consequences Of Martin's Axiom. Cambridge University Press (1984).

[23] D. H. Fremlin. Mesrîre Algebras, in Handbook of Boolean AJgebras (J. D. Monk, ed.). North-Holland, Amsterdam 1989.

[24] D. H. Fremlin. Problem List, Mach 1998.

[25] F. Galvin. Persond communication with S. TodorieviC.

[26] F. Galvin, S. Shelah. Some Gounterexamples In The Partition Calculus. J. Combinatorial Theory Ser. A 15 (1973) 167-174.

[27] J. Ginsburg. S-Spaces In Countably Compact Spaces Using Ostaszewski 's Method. Pacfic Journal of Math. 68, 2 (1977) 393-397.

[28] G. Gruenhage. Cosmicity of Cornetrizable Spaces. Trans. Amer. Math. Soc. 313, 1 (1989) 301-315.

[29] G. Gruenhage, P. Nyikos. Nonnality in X2for compact X. Trans. Amer. Math. Soc. 340, 2 (1993) 563-586. [30] A. Hajnal. Some Results and Problems in Set Theory. Acta Math. Acad. SU. Hungar. 11 (1960) 277-298.

[31] F- Hausdorff. Die Graduiemng Nach Dem Endverlauf. Abhandun. Konig. Schsis. Gessellsch. Wissenschaften, Math.-Phys. Kl. 31 (1909) 296-334.

[32] E. Helly. über Mengen Konvezer K-er Mit Geneinschaftlichen Punken. Jber. Deutsch. Math. Verein 32 (1923) 175-176.

[33] A. Horn, A. Tarski. Measures on Boolean Algebras. Trans. Amer. Math. SOC.64 (1978) 467-497.

[34] T. Jech. Non-provability of Souslin's Hypothesis. Comment. Math. Univ. Carolinae 8 (1967) 291-305.

[35] M. Katëtov. Complete Nonnality in Cartesian Products. Fund. Math. 36 (1948) 271-274.

[36] K. Kunen. A Compact L-Space. Topology Appl. 12 (1981) 283-287.

[37] K. Kunen. Set Theory. Studies in Logic and the Foundations of Mathe- matics, 102. North-Holland (1983).

[38] D. Kurepa. Ensembles Ordonnés Et Ramifiés. Publ. Math. Univ. Bel- grade, 4 (1935) 1-138.

[39] D. Kurepa. Sur Les Relations D'Ordre. Bull. Internat. Acad. Yougoslave 32 (1939) 66-76. [40] R. Laver. Partition Relations For Uncountable Cardinals 5 2H~,in Vol- ume II of Finite and Infinite Sets (A. Hajnal, R. Rado, Vera T. S6s eds.). North-Holland (1975) 1029-1043.

[41] R. Laver. Random Reals and Souslin nees. Proc. Amer. Math. Soc. 100 (1987) 531-534.

[42] D. Maharam. On Homogeneous Measure Algebras. Proc. Nat. Acad. Sci. USA 28 (1942) 108-111.

[43] E. M. Miller. A Note on Souslin's Problem. Amer. J. Math. 65 (1943) 673-678.

[44] R. D. Mauldin (ed.). The Scottish Book. Birkhauser (1981).

[45] J . Tatch Moore. Continzlous Colokgs A ssociated With Certain Char- acteristics of the Continuum. Discrete Math. 214 (2000) 263-273.

[46] J. Tatch Moore. A Linearly Fzbered Souslinean Space Under MA. Topol- ogy Proc. (to appear).

[47] A. Ostaszewski. On Countably Compact, Perfectly Normal Spaces. J. London Math. Soc. 14, 2 (1976) 505-516.

[48] F. P. Ramsey. On a Problem of Fomal Logic. Proc. London Math. Soc. 30 (1930) 264-286.

[49] J. Roitman. Adding a Random or Cohen Real: Topological Consequences and the Effect of Martin's Axiorn. Fund. Math. 103 (1979), 47-60. [50] J- Roitman. A Refomulation of S and L. Proc. Amer. Math. Soc. 69, 2 (1978) 344348.

[51] F. Rothberger. Sur La Familles Indenombrables De Suites De Nombres Naturels Et Les Problems Concernant La Propriete C. Proc. Cambridge Phil. Soc. 37 (1941) 109-126.

[52] W. Siexphiski. Sur Un Probléme De La Théorie Des Relation. Am. Scuola Norm. Sup. Pisa 2 (1933) 285-287.

[53] W.Sierpfiki. Sur Un Probléme De La Théorie Général Des Ensembles Équivalent Au Problème De Souslin. Fund. Math. 35 (1948) 165-174.

[54] N. A. Shanin. On Products of Topological Spaces. Tnidy Math. Inst. Akad. Nauk SSSR 24 (1948) 112.

[55] B. Shapirovskuu. Maps Onto Tychonofl Cubes. Russian Math. Surveys 53, 3 (1980) 145-153.

[56] S. Shelah. Proper and Improper Forcing. Springer-Verlag (1998).

[571 R. Solovay, S. Tennenbaum. Iterated Cohen Extensions and Souslin's Problem. Ann. of Math. 94 (1971), 201-245.

[58] M. Soush. Probléme S. hd.Math. 1 (1920) 233.

[59] M. H. Stone. The Theory of Representations for Boolean Algebras. 'Itans. Amer. Math. Soc. 39 (1936) 37-111. [60] 2. Szentmikl6ssy- S Spaces and L Spaces Under Martin's Axiom, in Vol- ume II of Topology. (Proc. Fourth CoUoq., Budapest, 1978). Cou. Math.

Soc. Janos Bolyai 23 North-Bolland (1980) 1139-1145

[61] A. Szymariski. Undecidability of the Existence of Regular Extremally Dis- connected S-Spaces. CoUoq. Math. 43 (1980) 61-67.

[62] F. Tall. The Density Topology. Pacific Journal of Math. 62, 1 (1976) 275-284.

[63] S. Tennenbaum. Souslin's Problem. Proc. Nat. Acad. Sci. 59 (1968) 60- 63.

[64] S. TodorCeviC. Analytic Gaps. Fùnd. Math. 150 (1996) 55-66.

[65] S. TodorEeviC. Cardinal Functions on Linearly Ordered Topologicd Spaces, in Topology and Ordered Structures Part 1(H. R. Bennett and D. J. Lutzer, eds.). Math Centrum, Tract 142, Amsterdam (1981) 177-

[66] S. TodorCeviC. Chain Condition Methods in Topology. Topology Appl. 101 (2000) 45-82.

[67] S. TodorEevit. The Gountable Chain Condition In Partition Calculus. Discrete Math. 188, 1-3 (1998) 205-223.

[68] S. TodorEeviE. Comparing the Continuum With the FiTst Two Uncount-

able Cardinals, in Logic and Scient& Methods (M. L. Dalla, et al, eds.). Kluwer Acad. Publ. (1997) 145-155. [69] S. TodorCevik. Definable Ideals And Gaps In Their Quotients, in Set Theory: Techniques And Applications (C. A. Di Prisco et al, eds.). Kluwer Acad. Press 1997, pp. 213-226.

[70] S. TodorCeviC. Forcing Positive Partition Relations. Trans. Amer. Math. SOC.280, 2 (1983) 703-720.

[71] S. TodorEeviC. Gaps in Analytic Quotients. Fund. Math. 156 (1998) 85-97.

[72] S. TodorCeviC. Oscillations of Real Num bers. Logic Colloquium '86. North-Holland (1998) 325-331.

[73] S. TodorCeviC. Partitioning Pairs Of Countable Ordinals. Acta Math. 159, 3-4 (1987) 261-294.

[74] S. TodorCeviC. Partition Problerns In Topology. American Math. Soc., Providence (1989).

[75] S. TodorCeviC. Random Set Mappings and Separability of Compacta. Topology and its Applications 74 (1996) 265-274.

[76] S. TodorEeviC Remarks On Cellularity In Products. Composito Math. 57 (1986) 357-372.

[77] S. TodorEeviC. Remarks On Chain Conditions In Products. Composito Math. 55, 3 (1985) 295-302.

[78] S. TodoreeviC. Remarks On Mahin's Anom And The Continuum Hy- pothesis. Can. J. Math. 43, 4 (1991) 832-851. (791 S. Todorcevic, 1. Farah. Some Applications of The Method of Forcing. Yenisei Series in Pure and Applied Mathematics. Yenisei, Moscow (1995).

[80] S. TodorCevE. Some Cornpactifieations of the Integers. Math. Proc. Cambridge Phil. Soc. 112 (1992) 247-254.

[81] S. TodoreeviC. Some Partitions Of Three-Dimensional Combinatorial Cubes. Journal of Combinatorial Set Theory 68, 2 (1994) 410-437.

[82] S. TodorCeviC. nees and Linearly Ordered Sets, in Handbook of

Set-Theoretic Topolo~(K. Kunen, J. Vaughan eds.) . North-Holland (1984).

[83] S. TodorCeviE, B. VeliCkoviC. Martin's Axiom and Partitions. CompositO Math. 63, 3 (1987) 391-408.

[84] L. B. Treybig. Conceming continuozls images of compact ordered spaces. Proc. Amer. Math- Soc. 15 (1964) 866-871.

[85] J. Vaughan. Small Uncountable Cardinals And Topology, in Open

Problems In Topology (J. van Mill, G. M. Reed eh.). North Holland (1990) 197-216.

[86] B. VelEkoviE. Forcing Axioms and Stationary Sets. Advances in Math. 94, 2 (1992) 256-284.

[87] M. L. Wage. Almost Disjoint Sets and Martin's Axïom. Journal Symbolic Logic 44, 3 (1979) 313-318. [88] M. L. Wage. Extremally Disconnected S-Spaces. Topology Proceedings, Vol 1 (Conf., Auburn Univ., 1976), 181-185.

[89] E. H. White. Topological Spaces in WchBlumburg's Theorem Holds. Proc. Amer. Math. Soc. 44 (1974) 315-339.

[go] J- Zapletal. Keeping Additivity Of The Nidl Ideal Small. Proc. Amer.

Math. Soc. 125, 8 (1997) 2443-2451. Index

W),74 otp(T), 69 TT,68 TF,69 c-centered, 7, 16, 21 a-linked, 12, 21, 35 ua, 95 a Lz b, 13 a sZ b, 13 tAi,95 a: <* y, 95 a: <" y, 95

alternation map, 40, 48, 49

Baire category, 4, 31 Barnett, J. H., 65, 67 Boolean algebra, 6, 12, 35

Cantor, G.,1 cardinal invariant INDEX

add(~V),35 add(N.), 42, 43 Dedekind, 1 add(N,,Ns), 42 density topology, 33, 86 add*(&), 22 discrete, 33, 82, 85 6, 12, 27, 40, 87, 94

c, 94 Eisworth, T., 90 non(Af), 32 Erdos, P., 43 centered, 6, 21 Farah, I., 23 chah condition filter, 6 global, 3, 4, 10 hee sequence, 90 local, 3, 5, 30 Fkemliil, D. H., 62, 92 Cohen, P., 8 coloring, 39 gap, 11, 12, 16, 17, 19, 22, 26 associated with analytic, 22 add(N,), 40, 41, 43 definable, 14 6, 40 linear, 19 continuous, 39, 40 Ginsburg, J., 87 irreducible, 39 graph, 32 ..Sierpiiiski, 39 countably chromatic, 32 used in coding, 40 Gruenhage, G., 83, 89

Continuum Hypothesis CH, 8, 66, Hajnal, A., 66, 67 90 partition under CH, 66 countable chah condition, 7 Helly, E-,20 cover, 42 condition, 20, 27, 29 theorem, 2, 20 linear compactum, 10 Hom, A., 3 Iinear order, 1, 2 linearly fibered, 11, 19, 22, 27, 29, ideal, 13, 21, 24 92 analytic, 14, 22, 24 Wed, 2, 6, 21 dense, 13, 24 P-ideal, 13, 14, 22, 24 Maharam, D., 62 theorem, 62 Jech, T., 4, 8 Martin's Axiom, 4, 5, 8, 56, 65 Katëtov, M-, 89 MAN,,4, 5, 8, 59, 65, 66, 72, Knaster, B., 2, 10 73, 83 condition, 2, 3, 5, 31, 80 Martin's Axiorn MAN,,87 in products, 3 Martin, D. A., 8, 30 condition K,, 31, 35 measure 0, 30, 33, 35, 40, 41 Kunen, K., 5, 65, 80, 83 measure algebra, 56, 57, 62 Kurepa, D., 2 character, 57, 62 factor, 57 L space, 83, 89 homogeneous, 58, 62 cornetrizable, 81, 83, 85, 86 measure preserving homomor- compact, 81, 83, 89, 90 phism, 62 compactifications, 92 projection, 58 extremally disconnected, 86 separable, 57 ladder system, 73 ultrapower, 58 Laver, R., 59, 65-67, 84

Lindelof, 82 Nyikos, P., 89 Open Coloring barn OCA, 28, Rowbottom, 5 40, 52, 53 S space, 83, 89 orthogonal, 13 cornetrizable, 83, 85-87 oscillation map, 40, 48, 49 compact, 83, 89, 90 Ostaszewski space, 89, 90 extremally disconnected, 83, 87 partial order, 6 Scottish Book, 2, 3 partition calculus, 66 separable, 1, 3, 4, 7, 12, 16, 22, 27, partition tree, 67 29, 82 TT,68 Shanh, N. A., 3 filter on, 68 condition, 3, 5, 18, 22, 31 order type, 69 in products, 3 tree-like, 69 Shapirovskiï, B., 11 ultrafilter, 69 Sierpiiiski, W., x uniforrn, 69 partition, x, 39, 66 Proper Forcing Axiom PFA, 4, 9, set, 67, 80, 86, 92 22, 40, 52, 53, 72, 83 Solovay, R., 4, 5, 8, 30

SousLin, M., X, 1 Q set, 89 condition, 1-5, 7, 10, 12, 17, Ramsey, F. P., x, 83 18, 22, 27, 29, 31, 65, 80 numbers, 40, 43 in products, 3, 80 theorem, x, 39, 66 line, 5, 13, 84 theory, 7 problem, x, 1, 2, 4, 8, 30 random graph, 56, 59 program, 4-6 Roitman, J., 65, 90 INDEX

tree, 65 split, 13 Stone, M. H., 6 representation t heorem, 6 space, 6, 12, 86 SzentmiklOssy, 83 Szpilrajn, E., 2, 10 Szymariski, A., 87, 88

Tarski, A., 3 Tennenbaurn, S., 4, 8, 30 TodorCeviE, 83 TodorCeviC, S., 5, 22, 27, 31, 32, 48, 49, 67, 81, 87, 90, 92 tree, 31, 32, 59 Treybig, L. B., 10 Tychonoff cube, 4, 10

Veliekovie, B., 5, 31 von Neumann's problern, 3

Wage, M., 77, 87 W(O), 77, 83, 87 Ward, A. J., 10 White, E. H., 86