tordorcevic - walks on ordinals

Progress in MathematicsVolume 263

Series Editors

H.BassJ. OesterlA. Weinstein

Stevo Todorcevic

Walks on Ordinals and Their Characteristics

BirkhuserBasel Boston Berlin

2000 Mathematics Subject Classication 03E10, 03E75, 05D10, 06A07, 46B03, 54D65, 54A25

Library of Congress Control Number: 2007933914

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Biblio-thek lists this publication in the Deutsche Nationalbibliograe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8528-6 Birkhuser Verlag AG, Basel Boston Berlin

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ISBN 978-3-7643-8528-6 e-ISBN 978-3-7643-8529-3

9 8 7 6 5 4 3 2 1 www.birkhauser.ch

Stevo TodorcevicUniversit Paris VII C.N.R.S. Department of MathematicsUMR 7056 University of Toronto2, Place Jussieu Case 7012 Toronto M5S 2E475251 Paris Cedex 05 CanadaFrance e-mail: [email protected]: [email protected]

and

Mathematical Institute, SANUKneza Mihaila 3511000 BelgradSerbiae-mail: [email protected]

Contents

1 Introduction

1.1 Walks and the metric theory of ordinals . . . . . . . . . . . . . . . 11.2 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Prerequisites and notation . . . . . . . . . . . . . . . . . . . . . . . 171.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Walks on Countable Ordinals

2.1 Walks on countable ordinals and their basic characteristics . . . . . 192.2 The coherence of maximal weights . . . . . . . . . . . . . . . . . . 292.3 Oscillations of traces . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 The number of steps and the last step functions . . . . . . . . . . . 47

3 Metric Theory of Countable Ordinals

3.1 Triangle inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Constructing a Souslin tree using . . . . . . . . . . . . . . . . . . 583.3 A Hausdor gap from . . . . . . . . . . . . . . . . . . . . . . . . 633.4 A general theory of subadditive functions on 1 . . . . . . . . . . 663.5 Conditional weakly null sequences based on

subadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Coherent Mappings and Trees

4.1 Coherent mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Lipschitz property of coherent trees . . . . . . . . . . . . . . . . . . 954.3 The global structure of the class of coherent trees . . . . . . . . . . 1084.4 Lexicographically ordered coherent trees . . . . . . . . . . . . . . . 1244.5 Stationary C-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 The Square-bracket Operation on Countable Ordinals

5.1 The upper trace and the square-bracket operation . . . . . . . . . 1335.2 Projecting the square-bracket operation . . . . . . . . . . . . . . . 1395.3 Some geometrical applications of the

square-bracket operation . . . . . . . . . . . . . . . . . . . . . . . . 144

vi Contents

5.4 A square-bracket operation from a special Aronszajn tree . . . . . 1525.5 A square-bracket operation from the complete binary tree . . . . . 157

6 General Walks and Their Characteristics

6.1 The full code and its application incharacterizing Mahlo cardinals . . . . . . . . . . . . . . . . . . . . 161

6.2 The weight function and its local versions . . . . . . . . . . . . . . 1746.3 Unboundedness of the number of steps . . . . . . . . . . . . . . . . 178

7 Square Sequences

7.1 Square sequences and their full lower traces . . . . . . . . . . . . . 1877.2 Square sequences and local versions of . . . . . . . . . . . . . . . 1957.3 Special square sequence and the corresponding function . . . . . 2027.4 The function on successors of regular cardinals . . . . . . . . . . 2057.5 Forcing constructions based on . . . . . . . . . . . . . . . . . . . 2137.6 The function on successors of singular cardinals . . . . . . . . . . 220

8 The Oscillation Mapping and the Square-bracket Operation

8.1 The oscillation mapping . . . . . . . . . . . . . . . . . . . . . . . . 2338.2 The trace lter and the square-bracket operation . . . . . . . . . . 2438.3 Projections of the square-bracket operation

on accessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 2518.4 Two more variations on the square-bracket operation . . . . . . . . 257

9 Unbounded Functions

9.1 Partial square-sequences . . . . . . . . . . . . . . . . . . . . . . . . 2719.2 Unbounded subadditive functions . . . . . . . . . . . . . . . . . . . 2739.3 Changs conjecture and 2 . . . . . . . . . . . . . . . . . . . . . . 2779.4 Higher dimensions and the continuum hypothesis . . . . . . . . . . 283

10 Higher Dimensions

10.1 Stepping-up to higher dimensions . . . . . . . . . . . . . . . . . . . 28910.2 Changs conjecture as a 3-dimensional

Ramsey-theoretic statement . . . . . . . . . . . . . . . . . . . . . . 29410.3 Three-dimensional oscillation mapping . . . . . . . . . . . . . . . . 29810.4 Two-cardinal walks . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Chapter 1

Introduction

1.1 Walks and the metric theory of ordinals

This book is devoted to a particular recursive method of constructing mathemati-cal structures that live on a given ordinal , using a single transformation Cwhich assigns to every ordinal < a set C of smaller ordinals that is closed andunbounded in the set of ordinals < . The transnite sequence

C ( < )

which we call a C-sequence and on which we base our recursive constructions mayhave a number of coherence properties and we shall give a detailed study of themand the way they inuence these constructions. Here, coherence usually meansthat the Cs are chosen in some canonical way, beyond the already mentioned andnatural requirement that C is closed and unbounded in for all . For example,choosing a canonical fundamental sequence of sets C for < 0, relyingon the specic properties of the Cantor normal form for ordinals below the rstordinal satisfying the equation x = x, is a basis for a number of importantresults in proof theory. In set theory, one is interested in longer sequences as welland usually has a dierent perspective in applications, so one is naturally led touse some other tools besides the Cantor normal form. It turns out that the setsC can not only be used as ladders for climbing up in recursive constructions butalso as tools for walking from an ordinal to a smaller one ,

= 0 > 1 > > n1 > n = ,where the step i i+1 is dened by letting i+1 be the minimal point of Cithat is bigger than or equal to . This notion of a walk and the correspondingcharacteristics and distance functions constitute the main body of study inthis book. We show that the resulting metric theory of ordinals is a theory ofconsiderable intrinsic interest which provides not only a unied approach to a

2 Chapter 1. Introduction

number of classical problems in set theory but is also easily applicable to otherareas of mathematics. For example, highly applicable characteristics of the walkare dened on the basis of the corresponding traces. The most natural trace ofthe walk is its upper trace dened simply to be the set

Tr(, ) = {0 > 1 > > n1 > n}of places visited along the way, which is of course most naturally enumerated indecreasing order. Another important trace of the walk is its lower trace, the set

(, ) = {0 1 n2 n1},

where i = max(i

j=0 Cj ) for i < n. The traces are usually used in deningvarious binary operations on ordinals < , the most prominent of which is thesquare-bracket operation that gives us a way to transfer the quantier for everyunbounded set to the quantier for every closed and unbounded set. It is perhapsnot surprising that this reduction of quantiers has proven to be quite useful inconstructions of mathematical structures on where one needs to have some gripon substructures of cardinality .

From the metric theory of ordinals based on analysis of walks, one also learnsthat the triangle inequality of an ultrametric

(, ) max{(, ), (, )}has three versions, depending on the natural ordering between the ordinals , and . The three versions of the inequality are in fact of a quite dierent characterand occur in quite dierent places and constructions in set theory. For example,the most frequent occurrence is the case < < , when the triangle inequalitybecomes something that one can call transitivity of . Considerably more subtle isthe case < < of this inequality1. It is this case of the inequality that capturesmost of the coherence properties found in this article. It is also an inequality thathas proven to be quite useful in applications.

A large portion of the book is organized as a discussion of four basic char-acteristics of the walk , 0, 1, 2 and 3. The reader may choose to follow theanalysis of any of these functions in various contexts. The characteristic 0(, )codes the entire walk = 0 > 1 > > n1 > n = by simply listing thepositions of i+1 in the set Ci for i < n. While this looks simple-minded, theresulting mapping 0 is a rather remarkable object. For example, in the realm ofthe space 1 of countable ordinals, it gives us a canonical example of a specialAronszajn tree of increasing sequences of rationals which has the additional re-markable property that, when ordered lexicographically, its cartesian square canbe covered by countably many chains. In other words, the single characteristic0 of walks on countable ordinals gives two critical structures, one in the class of1It appears that the third case < < of this inequality is rarely a reasonable assumption tobe made in this context.

1.1. Walks and the metric theory of ordinals 3

so-called Lipschitz trees and the other in the class of linear orderings. For highercardinals , analysis of 0 leads us to some interesting nitary characterizationsof hyper inaccessible cardinals. This is given in some detail in Chapter 6 of thisbook.

The characteristic 1(, ) loses a considerable amount of information aboutthe walk as it records only the maximal order type among the sets

{C0 ,C1 , . . . , Cn1 }.Nevertheless it gives us the rst example of what we call a coherent mapping. Theclass of coherent mappings and trees in the case = 1 exhibits an unexpectedstructure that we study in great detail in Chapter 4 of the book. The ne structurein the class of coherent trees is based on the metric notion of a Lipschitz mappingbetween trees. The profusion of such mappings between coherent trees eventuallyleads us to the so-called Lipschitz Map Conjecture that has proven crucial forthe nal resolution of the basis problem for uncountable linear orderings and thatis presented in the same chapter. For higher cardinals the characteristic 1 andits local versions oer a rich source of so-called unbounded functions that havesome applications.

The characteristic 2(, ) simply counts the number of steps of the walkfrom to . While this also looks rather simple minded, the remarkable proper-ties of the corresponding function 2 become especially apparent on higher car-dinals . Important properties of this characteristic are its coherence and its un-boundedness. The coherence property of 2 requires the corresponding C-sequenceC ( < ) to be coherent in the sense that C = C whenever is a limitpoint of C . On the other hand, the unboundedness of 2 translates into a require-ment that the corresponding C-sequence C ( < ) be nontrivial2, a conditionthat eventually leads us to a simple and natural characterization of weakly com-pact cardinals that we choose to reproduce in some detail in Chapter 6.

Finally, the characteristic 3(, ) attaches one of the digits 0 or 1 to the walkaccording to the behavior of the last step n1 n = . The full analysis ofthis characteristic is currently available only in the reals of the space 1, where 3becomes a rather canonical example of a sequence-coherent mapping with valuesin {0, 1} and with properties reminiscent of those appearing in the well-knownnotion of a Hausdor gap in the quotient algebra P()/n (another critical objectthat shows up in many problems about this quotient structure).

The true metric theory of ordinals comes only with development of thecharacteristic (, ) of the walk that takes advantage of the so-called full lowertrace of the walk. The depth of this characteristic is apparent even in the space 1of countable ordinals, but its full power comes at higher cardinals and especiallyat that are successors of singular cardinals. The full analysis of the characteristic requires C ( < ) to be a so-called square sequence or in other words requires2We say that C ( < ) is nontrivial if there is no closed and unbounded set C such that.for all limit points of C, there is such that C C .


the most widely known coherence condition on this sequence, which says that if is a limit point of C , then C = C. It is not surprising that this characteristichas the largest number of applications, many of which are reproduced in this book.We have already mentioned that its development in the case = 1 was the initialimpulse for development of the so-called metric theory of countable ordinals thathas already a rich spectrum of applications. At higher cardinals the characteristic can be used in facilitating set-theoretic forcing constructions of rather specialobjects and we shall reproduce some of these constructions in Chapter 7 of thisbook. While, as said above, the full development of requires C ( < ) to bea square sequence, the function itself holds considerable information aboutthe notion of square sequences. For example in Chapter 7, we use to turn anarbitrary square sequence C ( < ) into a non-special one by expressing theusual order relation among ordinals < as an increasing union of tree-orderingsthat come from square sequences on themselves. This again leads us to someapplications that we choose to reproduce in detail at the end of Chapter 7.

We have already mentioned that one of the important outcomes of our studyof walks on ordinals is the square-bracket operation, a transformation which toevery pair < of ordinals < assigns an ordinal [] belonging to the uppertrace Tr(, ) of the walk from to . We have also mentioned that the choice of[] has to be rather careful in order to reduce an arbitrary unbounded subset Aof to the corresponding set of values

{[] : , A and < }

that contains a closed and unbounded subset of relative to some xed stationaryset , which the C-sequence C ( < ) avoids3. We present several variationson the way [] is chosen, each of which works best in some particular context.The common feature of these denitions of [] is that they are all based on theoscillation mapping

osc : P()2 Carddened by

osc(x, y) = |x \ (sup(x y) + 1)/ |,where is the equivalence relation on x\ (sup(xy)+1) dened by letting if and only if the closed interval determined by the ordinals and contains nopoint from y. In other words, osc(x, y) is simply the number of convex pieces thatthe set x \ (sup(x y) + 1) is split into by the set y. The original theory of theoscillation mapping osc has been developed in the realm of partial functions from into . In other words, there is a well-developed theory of the oscillation mapping

osc(s, t) = |{ : s() t() but s(+) > t(+)}|, 43C ( < ) avoids if C = for all limit ordinals < .4Here, + is the immediate successor of in the common domain of s and t.


(see, for example, [111]), but the general theory works equally well and it will bein part reproduced here in Chapters 8 and 9. The common feature of all resultsof such a theory is identication of the notion of unbounded, in either of thetwo contexts, in such a way that the typical oscillation result would say thatthe set of values osc(x, y) the oscillation mapping takes when x and y run insidetwo unbounded sets is in some sense rich. In our context of dening the square-bracket operation, the sets x and y in osc(x, y) are members of our C-sequenceC ( < ) on which we base the notion of walk, and the notion of unboundedbecomes the familiar notion of nontriviality of C ( < ). This makes the square-bracket operation [] well dened in a wide variety of contexts and thereforequite applicable.

Judging from the applications found so far, it appears that in order to makea particular variation of the oscillation mapping or the square-bracket operationuseful, one needs to be be able to give a quite precise estimate of its behavior,not only on unbounded subsets of but also on families of pairwise-disjointnite subsets of . It is for this reason that denite results about any particularvariation of osc and [] presented in this book will typically be about familiesA of pairwise-disjoint nite subsets of . Given such a family A of cardinality, by going to a subfamily, we may assume that elements of A have some xednite cardinality n. So, given a in A one can view it enumerated increasingly asa(0), . . . , a(n 1). All variations of the square-bracket operation that we presentwill have the property that the set of ordinals < that can be represented as

= [a(0)b(0)] = [a(1)b(1)] = = [a(n 1)b(n 1)],

for some a < b in A, contains a closed and unbounded set relative to some xedstationary set which the C-sequence C ( < ) avoids. Many applicationshowever require that we know the values [a(i)b(j)] when i = j < n and when aand b run through A. It turns out that modulo taking a projection [[]] of [], (orin other words, modulo composing [] with a map from into ) for many of thesquare-bracket operations that we dene in this book, the particular set

{[[a(i)b(j)]] : i, j < n and i = j}

of values will be independent of the choice of a = b in A. This turns out to becrucial in several applications of [] presented in this book. Naturally, one wouldalso like to know whether one can dene a variation on [] where we would havefreedom of getting arbitrary values of the form [a(i)b(j)] independently of whetheri = j or not. It turns out that this is indeed possible for some choices of , thoughthe corresponding denitions are necessarily less general as they do not apply inthe case = 1 since otherwise one would be able to prove that the countablechain condition is not productive without appealing to additional axioms of settheory.5 In case = 1, we do have symmetric binary operations with some degree5Recall that the countable chain condition is a productive property under MA1 (see [36], 41E).


of freedom in that direction (see Chapter 2), but the exact breaking point betweenthis and what requires additional axioms of set theory has yet to be determined.

This book will also present some higher-dimensional characteristics of thewalk, though in that context the full theory is yet to be developed. For example,in Chapter 10, we consider the characteristic (, , ) which to any given threeordinals < < < assigns the place where the walk from to branches fromthe walk from to . It turns out that when C ( < +) is a square sequence,the characteristic can be used to step-up objects living on to objects on itssuccessor +. For example, one application of this characteristic is found in theproof that Changs Conjecture6 is equivalent to a 3-dimensional Ramsey-theoreticstatement saying that, for every coloring of [2]3 with 1 colors, there is an un-countable set B 2 which misses at least one of the colors. The 3-dimensionalcharacteristic (, , ) that simply measures the length of the common parts ofthe walks and can be used for detecting when a subset of admitsa rich restriction of the 3-dimensional version of the oscillation mapping,

osc : [+]3 ,dened on the basis of its 2-dimensional version as follows:

osc(, , ) = osc(Cs \ ,Ct \ ),where s = 0(, ) (, , ) and t = 0(, ) (, , ). Here s is the memberi of the trace of the walk = 0 > > l = whose code is the sequence s, i.e.,0(i, ) = s, and similarly t is the term j of the walk = 0 > > k = whose code is the sequence t. In Chapter 10 we show that our analysis of the3-dimensional version of the oscillation mapping leads naturally to a square-bracket operation in that dimension, though the full analogy is yet to be com-pleted as one still needs to determine the behavior of this operation on families ofpairwise-disjoint nite subsets of + (which at the moment seems elusive).

The nal section of Chapter 10 is concerned with generalizing the basic notionof walks to the context of sets of ordinals rather than ordinals themselves, withthe goal of obtaining two-cardinal versions of the square-bracket operation. Forexample, we show that for every pair of innite cardinals < with regular,there is a mapping c : [[]]2 such that, for every conal subset U of [] andevery , there exist x y in U such that c(x, y) = . Fuller analogues of thesquare-bracket operation are however obtained only in certain cases. For example,assuming that there is a stationary subset S of [] which is equinumerous witha locally countable7 subset of [] , one can dene a square-bracket operation

[]S : [[] ]2 []6Recall that Changs conjecture is the model-theoretic statement claiming that every model of acountable signature that has the form (2, 1,


with the property that, for every conal subset U of [], the set of all s S thatare not of the form [xy]S for some x y in U is not stationary in [] . Since underthe same assumption about the stationary set S, this set can be partitioned into |S|pairwise-disjoint stationary subsets, one gets the following: For every stationarysubset S of [] that is equinumerous with a locally countable subset of [], thereis a projection

[[..]]S : [[] ]2 S

of the square-bracket operation [..]S with the property that, for every conal setU [] and every z S, there exist x y in U such that [[xy]]S = z.

An interesting phenomenon that one realizes while analysing walks on or-dinals is the special role of the rst uncountable ordinal 1 in this theory. Anynatural coherency requirement on the sets C ( < ) that one nds in this theoryis satisable in the case = 1. How natural the notion of walk in this context iscan be seen from the fact that basically all of its characteristics lead us in one wayor the other to some critical structure that shows up in various rough classica-tions of mathematical structures. For example, any of the characteristics , 0, 1,2 and 3 of the walk that we study here lead us to the canonical linear orderingappearing on the list of ve linear orderings that forms a basis for the class of alluncountable linear orderings. The rst uncountable cardinal is the only cardinalon which the theory can be carried out without relying on additional axioms ofset theory. The rst uncountable cardinal is also the place where the theory hasits deepest applications as well as its most important open problems. This specialrole can perhaps be best explained by the fact that many set-theoretical problems,especially those coming from other elds of mathematics, are usually concernedonly about the duality between the countable and the uncountable rather thansome intricate relationship between two or more uncountable cardinalities. 8 Forexample, consider the classical problem coming from topology asking if the hered-itary separability of a given regular space X is equivalent to its dual requirementthat every collection of open subsets of X has a countable subcollection with thesame union.9 It turns out that this problem has a reformulation in terms of thebehavior of mappings of the form c : [1]2 2 on uncountable families A ofpairwise-disjoint nite subsets of 1. In other words, a mapping that has a cer-tain complex behavior on uncountable families of pairwise disjoint nite sheafswould lead to an example of a regular space in which one of the implications fails.For example, one can produce a hereditary separable non-Lindelof space assumingthere is c : [1]2 2 with the property that, for every uncountable family Aof pairwise-disjoint nite subsets of 1, all of some xed size n for every position

8This is of course not to say that an intricate relationship between two or more uncountablecardinalities may not be a protable detour in the course of solving such a problem. In fact, thisis one of the reasons for our attempt to develop the metric theory of ordinals without restrictingourselves to the realm of countable ordinals.9In other words, every subspace of X is Lindelof.


i0 < n and every requirement h : n 2, there exist a < b in A such that

c(a(i0), b(j)) = h(j) for all j < n.

On the other hand, there is a regular hereditary Lindelof non-separable space ifthere is c : [1]2 2 with the dual property that, for every uncountable familyA of pairwise-disjoint nite subsets of 1, all of some xed size n for every positionj0 < n and every requirement h : n 2, there exist a < b in A such that

c(a(i), b(j0)) = h(i) for all i < n.

It turns out that one cannot produce c having the rst property without appealingto additional axioms of set theory, but that a variation of the oscillation mappingthat we reproduce in Chapter 2 of this book will give us a c with the secondproperty and therefore a regular hereditary Lindelof non-separable space.

One can nd this kind of application of the oscillation mapping or the square-bracket operation on 1 in other areas of mathematics as well. For example con-sider the problem of nding a large subspace on which a given homogeneous poly-nomial P : X C is zero, where X is some Banach space over the eld C ofcomplex numbers.10 It is known that if X is innite-dimensional, then there willalways be an innite-dimensional subspace Y of X on which P is zero, but can onend larger Y assuming X is not separable? More precisely, if X is not separable,can one nd a non-separable subspace Y of X on which P vanishes? Note that acounterexample would be a polynomial P : X C which takes all the complexvalues on any non-separable subspace of X . So it is natural to try appealing tothe square-bracket operation for constructing such a polynomial P . It turns outthat this is indeed possible and we shall reproduce it in Chapter 5 of this bookfor the space X = 1(1). In order to apply the square-bracket operation we needto take one of its convenient projections [[]]G0 with only countably many values,which one may assume to form a countable dense subset D of the unit disc of thecomplex plane. Then the polynomial P : 1(1) C is dened as

P (x) =

[[]]G0xx .

In order to establish that P takes all the values from C on any closed non-separablesubspace of 1(1), one uses the following property of the projection [[]]G0 : Forevery uncountable family A of pairwise-disjoint nite subsets of 1, all of thesame size n, there is uncountable set B A, an equivalence relation on n ={0, 1, . . . , n 1}, and a single mapping h : n n D such that

[[a(i)b(j)]]G0 = h(i, j) for all a < b in B and i j.

10Recall that a homogeneous polynomial of degree n on a Banach space X is a mapping P :X C of the form P (x) = (x, x, . . . , x), where : Xn C is a bounded symmetric n-linearform on X.


Moreover, for every uncountable C B and for every g : n n D there exista < b in C such that

[[a(i)b(j)]]G0 = g(i, j) for all i, j < n with i j.This particular application makes it clear how useful it is to know behavior ofthe square-bracket operation not only on uncountable subsets of 1 but also onarbitrary uncountable families A of pairwise-disjoint subsets of 1.

Minimal walks in general and the metric theory of ordinals in particularare applicable to problems that are not necessarily problems about uncountablestructures but rather about the behavior of countable substructures of a xedlarge structure. We demonstrate this in Chapter 3 where we present some ap-plications of the metric theory of countable ordinals to classical problems frominnite-dimensional geometry, such as, for example, the distortion problem or theunconditional basic sequence problem. Given a function

: []2 that is locally nite11 and that satises the two ultrametric inequalities,12 onecan use it to dene special functionals on the vector space c00() of all nitelysupported maps from the ordinal into the reals. Let e < be the standardHamel basis of c00(). Let e < be the corresponding sequence of biorthogonalfunctionals. We say that a sequence (Ei)i


We have already noted that the largest ordinal supporting such a -function is therst uncountable ordinal 1. It follows therefore that one can have an uncountableweakly-null sequence with no innite unconditional subsequence. It follows thatthe function can be used to control all countably innite subsets of the longweakly-null sequence e ( < 1). With a considerable amount of additional workone can even build a reexive Banach space X1 with a Schauder basis of length1 with no innite unconditional basic sequence. Extending these ideas furtherone can use to build another Banach space X 01 with a Schauder basis of length1, which on one hand keeps the distortion and all the conditional structure of thespace X1 at the level of its nonseparable subspaces, but on the other hand, everyinnite-dimensional closed subspace of X 01 contains an isomorphic copy of thespace c0, or in other words separable subspaces of X 01 do not hold any conditionalstructure nor do they allow their norms to be distorted.

1.2 Summary of results

Many chapters of the book can be read independently from each other. Once thebasic denition of the walk is understood, the reader may choose to follow thedevelopment of a particular characteristic in various contexts. The reader inclinedtowards applications of the methods of this book may start by reading about aparticular application and go back towards background material that is needed.In this section, we include a short summary that might help the reader in ndingspecic results presented in this book.

The rst section of Chapter 2 is the one to be rst read as it presents thenotion of minimal walk along a given C-sequence and the corresponding charac-teristic 0 that codes it. The resulting tree T (0) is a tree of height 1 that hasall of its levels countable and which admits a natural strictly increasing map intothe rationals. This could be the shortest construction of such a tree found in theliterature and surely is the most canonical one. This could be seen for exampleon the basis of the fact that if we order T (0) lexicographically we obtain an un-countable linearly ordered set whose cartesian square can be covered by countablymany chains. The proof of this fact depends on the development of the full lowertrace of the minimal walk and the reader is advised to skip this on the rst readingand go instead to the second section of Chapter 2 which presents the characteristic1 and the corresponding tree T (1) for which this fact is easier to establish. Thecharacteristic 1 has its own interesting application presented in the same section.For example, we show that the tree T (1) naturally leads us to an example ofa homogeneous non metrizable compactum that can be represented as a weaklycompact subset of some Banach space. As another application of 1 presented inthe same section is a functor that transfers a given graph G on the vertex set 1to a graph G on the same vertex set such that if G has an uncountable cliquethen the vertex set 1 can be covered by countably many sets that are cliques ofG. Moreover, we show that G satises the countable chain condition provided G

1.2. Summary of results 11

satises a slightly stronger version of this condition saying that for every uncount-able family F of pairwise disjoint nite subsets of 1, we can nd a = b in F suchthat {, } is an edge of G for every a and b. The point of this is that es-sentially all known ccc graphs on 1 do satisfy this stronger form of the countablechain condition and therefore this in particular shows that many of the standardconsequences of MA1 are Ramsey-theoretic in nature. Recall that it is still notknown if MA1 is in fact equivalent to the statement that every ccc graph on thevertex set 1 contains an uncountable clique. We nish this section with a proofthat unlike T (0) the tree T (1) does not always admit a strictly increasing mapinto the rationals. This amounts to measuring how much information about theminimal walk is lost when one passes from the full code 0(, ) to the maximalweight 1(, ) of the walk.

The third section of Chapter 2 presents an important theme that is goingto be developed in later parts of the book, the theme of the oscillation mapping.More precisely, in Section 2.3, we present two oscillation mappings osc0 and osc1corresponding to the upper and lower trace of the minimal walk, respectively. Ofspecic applications of these two oscillation mappings, we present a rather abso-lute decomposition of 1 into innitely many pairwise disjoint stationary subsets,and an example of a regular hereditarily Lindelof topological space that is notseparable. In Section 2.3 we introduce two new characteristics 2 and 3. Thecharacteristic 2 while quite simple minded in that it counts only the number ofsteps of the minimal walk its full power will become apparent only in later partsof the book when dealing with walks on larger cardinals. In fact, as far as weknow, 2 could be the rst nontrivial two-place mapping from a large cardinal into in the sense that it takes arbitrarily high value from on any productof two unbounded subsets of . Recall that large cardinals typically have theRamsey-theoretic properties saying that maps f : []2 are constant on []2for large subsets . The mapping 3 while considerably more subtle makessense only in the realm of countable ordinals. To a given walk from a countableordinal down to a smaller ordinal , the characteristic 3 assigns one of thedigits 0 or 1 according to what happens on the last step of the walk. Since it isa coherent mapping it shares many properties with the characteristic 1 thoughwe shall show that, unlike T (1), if we order lexicographically T (3), we are notalways guaranteed that the cartesian square of the corresponding uncountable lin-ear ordering can be covered by countably many chains. On the other hand, it istrue that every uncountable subset X of T (3) contains an uncountable subset Ywhich when ordered lexicographically has the property that its cartesian squarecan be covered by countably many chains. This again amounts to measuring howmuch of the information is lost about the minimal walks by passing from the char-acteristic 1(, ) to the characteristic 3(, ). However, while 3 looses much ofthe information about the minimal walk, it still gives us a rather interesting andcanonical object on 1 that is very much reminiscent of the classical notion of aHausdor gap in the quotient algebra P()/n.


In Chapter 3 we develop the metric theory of countable ordinals concentratedaround the two ultrametric triangle inequalities

d(, ) max{d(, ), d(, )} and d(, ) max{d(, ), d(, )},

whenever < < . We have already mentioned two applications of this theoryfound in this chapter, a reexive Banach space X1 with a Schauder basis of length1 with no innite unconditional basic sequence, and another Banach space X 01with a Schauder basis of length 1 that is, on one hand saturated with copies ofthe space c0 but on the other hand its norm admits an arbitrarily high distortionrelative to the class of nonseparable subspaces of X 01 . Chapter 3 presents alsothe characteristic of the minimal walk which besides the two mentioned triangleinequalities has many other interesting properties that show their full power onlarger cardinals than 1. In Chapter 3 we do present two important objects thatare naturally derived from , the Cohen name for a Souslin tree and a particularlycanonical example of a Hausdor gap in P()/n. The Souslin tree has domain 1and the property that no innite ground model set can be its chain or antichain.In Section 3.4 we study the general theory of functions : [1]2 that satisfythe two ultrametric triangle inequalities and that are locally nite in the sets thatthe set { < : (, ) n} is nite for every < 1 and every n < . We showthat there is a vast variety of such mappings including the universal one whoseconstruction we reproduce in the same section.

In Chapter 4 we develop a metric theory of trees that lies behind the prop-erties of the tree T (1). The basic notion here is that of a Lipschitz map betweentrees, a level preserving map with the property that (g(x), g(y)) (x, y) for allx and y in its domain. This leads us to the notion of a Lipschitz tree, an uncount-able tree T with the property that every level preserving map from an uncountablesubset of T into T has a Lipschitz restriction on some uncountable subset of itsdomain. The main purpose of Chapter 4 is to study the class C of Lipschitz treesas a structure equipped with the quasi ordering S T dened to hold wheneverthere is a Lipschitz map from the tree S into the tree T , or equivalently wheneverthere is a strictly increasing map from S into T. We also examine the relationshipbetween the class C and the larger class A of Aronszajn trees. Most of the theoryis developed without appeal to additional axioms of set theory though in the sameplaces we have used either MA1 or the Proper Forcing Axiom. It turns out thatLipschitz trees share many of the properties of the tree T (1) such as for examplethe property that when ordered lexicographically the cartesian square of the corre-sponding uncountable linear ordering can be covered by countably many chains. Itturns out that all trees T (0), T (1), T (2), T (3), and T () associated to variouscharacteristics of walks in 1 are Lipschitz. It turns out also that the class C istotally ordered under . In fact the chain (C,) is discrete in the sense that everyT in C admits a naturally dened shift T (1) that forms an immediate successorof T in C. In fact, the shift T (1) is also an immediate successor of T even in thebigger class A and this is essentially equivalent to Shelahs Conjecture saying that


an uncountable ordering either contains an uncountable well ordered or converselywell ordered subset, an uncountable separable ordered subset, or an uncountablesubset whose cartesian square can be covered by countably many chains. It turnsout that the chain C is not well ordered and this in particular solves an old problemof R. Laver asking if the class A is well quasi-ordered under a stronger orderingthan . Chapter 4 contains many other structural results about the classes C andA. For example, we show that while A is not totally ordered under , the chainC is both conal and coinitial in (A,).

In Chapter 5 we introduce and study the square-bracket operation [] forpairs and of countable ordinals. The ordinal [] is taken from the upper traceof the minimal walk from to . In later parts of the book we shall see severalvariation of this choice of [] but the basic idea is always the same and it is basedon the oscillation mapping osc0 of upper traces exposed above in Section 2.3. InChapter 5 itself we present two other variations on the square-bracket operation,one based on special Aronszajn tree and the other on the tree of all nite binarysequences. What the square-bracket operation adds to the space 1 of countableordinals is a rigidity which can formally be expressed by the fact that there is asentence of L(Q2) which has only rigid models. This is presented in Example 5.1.10which solves a problem of Ebbinghaus and Flum who showed that every modelof a sentence of L(Q) has a nontrivial automorphism. In Section 5.3 we presenttree geometrical application of the square-bracket operation. The rst one is anexample of a projective geometry of points and hyperplanes in Rn. The secondone is the example of a complex polynomial on (1) with no nonseparable nullsubspace already mentioned above. The third application of [] is an example ofa reexive Banach space X with a Schauder basis of length 1 in which everyoperator T can be written as I +S, where S is an operator with separable range.While in some sense this example is subsumed by the example of the Banach spaceX1 from Chapter 3 we have included it as its construction uses a quite dierentset of ideas. In Section 5.4 we give a formal explanation of how the square-bracketoperation reduces the quantication over uncountable subsets of 1 to that overclosed and unbounded subsets of 1. Answering a question of W.H. Woodin, weshow that there is a natural functor based on [] which to every subset on 1associates a graph K [1]2 such that, modulo PFA or Woodins axiom (), asubset of 1 contains a closed and unbounded set if and only if K contains[X ]2 for some uncountable X 1.

In Chapter 6 we develop the characteristic 0 of the minimal walk in thegeneral context. Recall that in the context of walks on 1 the tree T (0) that cor-responds to the characteristics 0 is special in the sense that it can be decomposedinto countably many antichains. In Chapter 6 we show that a strongly inaccessiblecardinal is not Mahlo precisely when one is able to nd a C-sequence in forwhich the corresponding tree T (0) special in the sense that it admits a regressivemapping that is not constant on any subset of T (0) that cannot be covered by lessthan antichains. The idea is then used in providing a unied approach towards


Ramsey-theoretic characterizations of n-Mahlo cardinals in terms of the existenceof min-homogeneous sets relative to regressive maps of the form f : []n+2 or f : []n+3 . In the same chapter we develop the general theory of thecharacteristics 1 and 2. In particular, in Section 6.2 we develop the local version1 : []

2 of 1 for an arbitrary regular cardinal < . The main interest inthis variation of 1 is that under some very mild assumptions about one obtainsmappings from []2 into that have strong unboundedness properties. Similar un-boundedness property of the characteristic 2 : []2 is equivalent to the nontriviality property of the C-sequence C ( < ) saying that there is no closedand unbounded subset C of such that for every < there is such thatC C . This gives a vast variety of cardinals for which one can nd aC-sequence C ( < ) such that the corresponding function 2 : []2 isstrongly unbounded. This can be seen from the characterization of weakly com-pact cardinals given in Section 6.3 which says that a strongly inaccessible cardinal is weakly compact if and only if for every C-sequence C ( < ) there is aclosed and unbounded set C such that for all < there is such thatC = C . We nish Chapter 6 with a particular topological application ofthe unboundedness property of 2

In Chapter 7 we study walks based on C-sequences C ( < ) that satisfythe coherence property saying that C = C whenever is a limit point of C .We call such C-sequences square sequences. It is this condition on the C-sequencewhich permits that the full theory of walks on 1 be lifted to the level thatsupports it. For example, we show that the full lower trace F(, ) of walks alongsquare sequence keeps all its properties from the context of countable ordinals. Asan application we show that the characteristic 2 in this context has an interestingcoherence property saying that sup


of the corresponding characteristic and the derived set-mapping D : [+]2 [+]


square has cellularity > . In fact, this result is true for arbitrary uncountable car-dinal that is not necessarily regular. This is proved in a similar fashion usingthe colorings dened on successors of singular cardinals exposed in Chapter 9 ofthis book.

In the rst section of Chapter 9, we use the characteristic in producing par-tial square sequence that can sometimes be used in places of full square sequences.In the second section of Chapter 9, we show that for a regular uncountable cardinal, there is a structure of the form (+, , Rn)n

1.3. Prerequisites and notation 17

a further elaboration. For example, we show that if there is a stationary subsetof [] that is equinumerous with a locally countable subset of [] then there isa variation of the square-bracket operation []S : [[] ]2 [] with propertiesquite analogous to those of the square-bracket operation of 1 (which is really thecase = and = 1 ) in the sense that for every conal subset U of [] theset of all s S that are not of the form [xy]S for x y in U is not stationaryin []. One of the points of this variation of the square bracket operation is thatthe set S can be split into |S| pairwise disjoint stationary sets, and since quitefrequently S has cardinality that is bigger than , we can dene projections[[]]S : [[]]2 that take more than colors (more precisely, = |S| colors)on any conal subset U of [] .

1.3 Prerequisites and notation

We have tried to keep the prerequisites needed for mastering the material of thisbook to a minimum. Though no specic training in set theory is necessary, thereader should be familiar with the notion of ordinal and recursive denitions andinductive proofs on them. This all can be found in most of the introductory textson the subject such as, for example, the newer ones [51], [64] and [55], or the oldertext [62] which has particularly detailed expositions of the recursive denitions overordinals. By comparing, the reader will notice that we are using standard notationin essentially complete agreement with these textbooks. In these sources the readerwill nd all the operations and properties of ordinals that we will use, but if morecomplete treatment is needed the reader may also consult one of the specializedtexts like [7]. We shall also look at ordinals with their natural topology induced byorder. It is in this context that we refer to closed subsets of a particular ordinal. The notions of unbounded and conality in this context refer of course to thenatural ordering of . The combinatorially inclined readers will notice that we areadopting the ErdosRado notation for the symmetric powers [S], the collectionsof all subsets of the set S of cardinality . Of special interest are of course thenite symmetric powers [S]k and the mappings dened on them because of theclear connection that this book has with Ramsey theory (see [40], [33] and [133]).In fact most of our characteristics of walks are mappings with domains equal tosymmetric squares []2 of some ordinals . A pair {, } []2 is usually assumedto be written such that < . In other words, sometimes it is convenient toidentify the symmetric square []2 with the set

{(, ) 2 : < }

of ordered pairs. In this way a given characteristic f : []2 of the walk canbe identied with the sequence f : ( < ) of ber mappings dened by

f() = f(, ).


Also we can more clearly express when the given characteristic f is coherent insome way by referring to the coherence between the corresponding ber mappings.Moreover, this identication allows us to use the notation f(, ) instead of themore cumbersome notation f({, }). Basically all our characteristics

f : []2

of walks are dened recursively in the sense that f(, ) is dened on the basisof the values f(, ) on lexicographically smaller pairs (, ). All the recursivedenitions require us to specify the boundary values f(, ) which are typicallytaken to be constant values such us 0 or depending on the context. It is for thisreason that sometimes we implicitly assume that the diagonal {(, ) : < } isa part of the domain of f .

1.4 Acknowledgements

We are grateful to a large number of people who have read versions of the manu-script and suggested corrections and improvements or helped us with preparationof the nal manuscript. Of these we would like in particular to mention MaximBurke, Christine Hartl, Akihiro Kanamori, Bernhard Konig, Piotr Koszmider, Car-los Martinez, Justin Moore and Luis Pereira. Special thanks are due to JordiLopez-Abad for helping us in designing the diagrams and gures of the book.

Chapter 2

Walks on Countable Ordinals

2.1 Walks on countable ordinals andtheir basic characteristics

The space 1 of countable ordinals is by far the most interesting space consideredin this book. There are many mathematical problems whose combinatorial essencecan be reformulated as problems about 1, which is in some sense the smallestuncountable structure. What we mean by structure is 1 together with a systemC ( < 1) of fundamental sequences, i.e., a system with the following twoproperties:

(a) C+1 = {},(b) C is an unbounded subset of of order-type , whenever is a countable

limit ordinal > 0.

Despite its simplicity, this structure can be used to derive virtually all knownother structures that have been dened so far on 1. There is a natural recursiveway of picking up the fundamental sequences C, a recursion that refers to the Can-tor normal form which works well for, say, ordinals < 01. For longer fundamentalsequences one typically relies on some other principles of recursive denitions andone typically works with fundamental sequences with as few extra properties aspossible. We shall see that the following assumption is what is frequently neededand will therefore be implicitly assumed whenever necessary:

(c) if is a limit ordinal, then C does not contain limit ordinals.

1One is tempted to believe that the recursion can be stretched all the way up to 1 and this isprobably the way P.S. Alexandro discovered the phenomenon that regressive mappings on 1must be constant on uncountable sets (see [2] and [3, Appendix]).

20 Chapter 2. Walks on Countable Ordinals

Denition 2.1.1. A step from a countable ordinal towards a smaller ordinal isthe minimal point of C that is . The cardinality of the set C , or betterto say the order-type of this set, is the weight of the step.

Denition 2.1.2. A walk (or a minimal walk) from a countable ordinal to asmaller ordinal is the sequence = 0 > 1 > > n = such that for eachi < n, the ordinal i+1 is the step from i towards .

Analysis of this notion leads to several two-place functions on 1 that havea rich structure and many applications. Let us expose some of these functions.

Denition 2.1.3. The full code of the walk is the function 0 : [1]2

2.1. Walks on countable ordinals and their basic characteristics 21

Figure 2.1: The walk and its traces.

Thus, if = 0 > 1 > > n = is the minimal walk from to (i.e.,0 = , n = , and i+1 = min(Ci \ ) for i < n), then

Tr(, ) = {i : 0 i n} and L(, ) = {i : 0 i < n},where i = max(

ij=0 Cj ). The following immediate facts about these notions

will be frequently and often implicitly used.

Lemma 2.1.6. For ,(a) > L(, ) implies that 0(, ) = 0(, )0(, ) and therefore that

Tr(, ) = Tr(, ) Tr(, ),(b) L(, ) > L(, ) implies that L(, ) = L(, ) L(, ).4

4For two sets of ordinals F and G, by F < G we denote the fact that every ordinal from F issmaller than every ordinal from G. When G = {}, we write F < rather than F < {}.


Denition 2.1.7. The full lower trace of the minimal walk is the functionF : [1]2 [1]


Proof. The proof is again by induction on . Let 1 = min(F(, ) \ ) and1 = min(C \ ). Let 1 be the minimal (C ) {min(C \ )} such that1 F(, ) (or F(, )).

Applying the inductive hypothesis to 1 (or 1) and notingthat 1 = min(F(1, ) \ ) (or 1 = min(F(, 1) \ )), we get

0(, ) = 0(1, )0(, 1), (2.1.1)

the part (a) of Lemma 2.1.9. Note that depending on whether 1 or 1, we have that F(1, ) F(, ) or F(, 1) F(, ) because in thelatter case 1 is also equal to min(C \ ). Dene

2 := min(F(1, ) \ ) or 2 := min(F(, 1) \ ),

depending on whether 1 or 1, respectively. Note that in anyof the cases of the denition of 2, we have that 2 1. Applying the inductivehypothesis to 1 (or 1) we get:

0(, ) = 0(2, )0(, 2), (2.1.2)

0(, 1) = 0(2, 1)0(, 2). (2.1.3)

Comparing (2.1.1) and (2.1.2) we see that 0(, 1) is a tail of 0(, 2), so theequation (2.1.3) can be rewritten as

0(, 1) = 0(2, 1)0(1, 2)0(, 1) = 0(1, 1)0(, 1). (2.1.4)

Finally since 0(, ) = |C |0(, 1) = 0(1, )0(, 1) holds, theequation (2.1.4) gives us the desired conclusion (b) of Lemma 2.1.9.

Denition 2.1.10. Recall that right lexicographical ordering on


Proof. It suces to decompose the set of all pairs (, ) where < . To each suchpair we associate a hereditarily nite set p(, ) which codes the nite structureobtained from F(, ) {} by adding relations that describe the way 0 acts onit. To show that this parametrization works, suppose we are given two pairs (, )and (, ) such that

p(, ) = p(, ) and


Theorem 2.1.12. Under MA1 , the ordering C(0) is a minimal uncountable linearordering in the sense that every linear ordering that embeds into C(0) but is notequivalent6 to it must be countable.

Proof. Let be a given uncountable subset of 1. We need to construct a mappingf : 1 that is strictly increasing relative to the ordering


for all i = 1, . . . , k, where (1), . . . , (k) and (1), . . . , (k) are increasingenumerations of dom(p) \ dom(r) and dom(p) \ dom(r), respectively. It shouldbe clear that this means that p p satises (3) and (4) as well.

If we choose the closed unbounded set D to be the trace on 1 of somecontinuous -chain of countable elementary submodels of (H(1),), one easilychecks that for all < 1, the set

D = {p P : dom(p)}

is a dense subset of P . So an application of MA1 gives us a mapping f : 1 that is strictly increasing relative to the ordering


of elements of L. So going to an uncountable subfamily of X0, we may assume that{p(1), . . . , p(k) : p X0} is a chain in C(0)k and that {f(p(1)), . . . , f(p(k)) :p X0} is a chain in Lk. Using 2.1.13 it follows now that f is decreasing onK = {p(1) : p X0}. Combining this with Theorem 2.1.12, we get the desiredconclusion C(0) K L of the corollary. Remark 2.1.14. It follows that, up to the equivalence, the ordering C(0) and itsreverse C(0) will appear in any basis9 of the class of uncountable linear orderingsas long as the axiom used in nding the basis includes the weak forms of MA1used in the proofs of Theorem 2.1.12 and Corollary 2.1.13. In fact, the combinationof the results of [10] and [77] shows that under the Proper Forcing Axiom,10 thelist

1, 1 , B, C(0), C(0)

(2.1.14)

form a basis for the class of all uncountable linear orderings, where B is any set ofreals of cardinality 1. We shall return to this subject matter in Section 4.4 below.Notation 2.1.15. Well-ordered sets of rationals. The set


Lemma 2.1.18. { < : 0(, ) = 0(, )} is a closed subset of whenever < .

Proof. Let < be a given accumulation point of this set and let = 0 > >n = and = 0 > > m = be the traces of the walks and ,respectively. Being an accumulation point of the set { < : 0(, ) = 0(, )},the ordinal is in particular a limit ordinal, so we can nd an ordinal < suchthat 0(, ) = 0(, ) and

> max(Ci ) and > max(Cj ) for all i < n and j < m. (2.1.15)

Note that this in particular means that the walk is a common tail ofthe walks and . Subtracting 0(, ) from 0(, ) we get 0(, )and subtracting 0(, ) from 0(, ) we get 0(, ). It follows that 0(, ) =0(, ).

It follows that T (0) does not branch at limit levels. Now note the generalfact that any downward closed subtree of Q which is nitely branching at limitnodes admits a strictly increasing mapping into Q, so the tree T (0) is a specialsubtree of Q.

Denition 2.1.19. Identifying the power set of Q with the particular copy 2Q ofthe Cantor set, dene for every countable ordinal ,

G = {x 2Q : x end-extends no 0 for }.

Lemma 2.1.20. G ( < 1) is an increasing sequence of proper G-subsets of theCantor set whose union is equal to the Cantor set.

Lemma 2.1.21. The set X = {0 : < 1} considered as a subset of the Cantorset 2Q has universal measure zero.

Proof. Let be a given nonatomic Borel measure on 2Q. For t T (0), set

Pt = {x 2Q : x end-extends t}.

Note that each Pt is a perfect subset of 2Q and therefore is -measurable. Let

S = {t T (0) : (Pt) > 0}.

Then S is a downward closed subtree of Q with no uncountable antichains. Byan old result of Kurepa (see [105]), no Souslin tree admits a strictly increasingmap into the reals (as for example Q does). It follows that S must be countableand so we are done.

2.2. The coherence of maximal weights 29

2.2 The coherence of maximal weights

Denition 2.2.1. The maximal weight of the walk is the characteristic

1 : [1]2 dened recursively by

1(, ) = max{|C |, 1(,min(C \ ))},with the boundary value 1(, ) = 0 for all . Thus 1(, ) is simply the maximalinteger appearing in the sequence 0(, ) dened above.

Lemma 2.2.2. For all < < 1 and n < ,

(a) { : 1(, ) n} is nite,(b) { : 1(, ) = 1(, )} is nite.Proof. The proof is by induction. To prove (a) it suces to show that for everyn < and every A of order-type there is A such that 1(, ) > n.Let = sup(A). If = one chooses arbitrary A with the property that|C | > n, so let us consider the case < . Let 1 = min(C \ ). By theinductive hypothesis there is A such that:

> max(C ), (2.2.1)1(, 1) > n. (2.2.2)

Note that 0(, ) = |C |0(, 1), and therefore1(, ) 1(, 1) > n.

To prove (b) we show by induction that for every A of order-type thereexists A such that 1(, ) = 1(, ). Let = sup(A) and let 1 = min(C\).Let n = |C | and let

B = { A : > max(C ) and 1(, 1) > n}.Then B is innite, so by the induction hypothesis we can nd B such that1(, ) = 1(, 1). Then

1(, ) = max{n, 1(, 1)} = 1(, 1),so we are done. Denition 2.2.3. Consider the following linear ordering


Lemma 2.2.4. The cartesian square of the total ordering


Lemma 2.2.6. For every pair A and B of uncountable families of pairwise-disjointnite subsets of 1 and every positive integer n, there exist uncountable subfamiliesA0 A and B0 B such that, for every pair a A0 and b B0 such that a < b,15we have 1(, ) > n for all a and b. Corollary 2.2.7. The tree

T (1) = {t : : < 1, t = 1}16

is a coherent, homogeneous, Aronszajn tree that admits a strictly increasing mapinto the reals. Denition 2.2.8. Consider the following extension of T (1):

T (1) = {t : : < 1 and t T (1) for all < }.If we order T (1) by the right lexicographical ordering 0, set

A(1) = {t A(1) : l(t) }.Then each A(1) is a closed subset of A(1) with the level T (1) being a count-able order-dense subset. One easily concludes from this that A(1) ( limit < 1)is the required decomposition of A(1).

To show that A(1) is homogeneous, consider two pairs x0 < x1 and y0 < y1of non-endpoints of A(1). Choose a countable limit ordinal strictly above thelengths of x0, x1, y0 and y1. So we have a Cantor set A(1), an order-dense subsetT (1) of A(1), and two pairs of points x0 < x1 and y0 < y1 in

A(1) \ T (1).By Cantors theorem there is an order isomorphism : A(1) A(1) suchthat:

T (1) = T (1), (2.2.5)

(xi) = yi for i < 2. (2.2.6)

15For sets of ordinals a and b we write a < b whenever < for all a and b.16Here, = denotes the fact that the functions agree on all but nitely many arguments.


Extend to the rest of A(1) by the formula

(t) = (t )t [, l(t)).

It is easily checked that the function is the required order-isomorphism.Let us now prove that there is no order-reversing bijection

: A(1) A(1).Since T (1) with the lexicographical ordering does not have a countable order-dense subset, the image T (1) must have sequences of length bigger than anygiven countable ordinal. So for every limit ordinal we can x t in T (1) oflength such that (t) also has length . Let

f() = max{ < : t() = (t)()} + 1.By the Pressing Down Lemma17, nd an uncountable set of countable limitordinals, a countable ordinal and s, t T

(1) such that for all :

f() = , (2.2.7)

t = s and (t) = t. (2.2.8)Moreover we may assume that

{t : } and {(t) : }are antichains of the tree T (1). Pick = in and suppose for deniteness thatt


the set of all downward closed chains of the tree T 0(1) and the topology on T (1)is simply the topology one obtains from identifying the power set of T 0(1) withthe cube

{0, 1}T 0(1)

with its Tychono topology18. T (1) being a closed subset of the cube is compact.In fact T (1) has some very strong topological properties such as the propertythat closed subsets of T (1) are its retracts.

Lemma 2.2.11. T (1) is a homogeneous Eberlein compactum19.

Proof. The proof that T (1) is homogeneous is quite similar to the correspondingpart of the proof of Lemma 2.2.9. To see that T (1) is an Eberlein compactum,i.e., that the function space C(T (1)) is weakly compactly generated, let {Xn} bea countable antichain decomposition of T 0(1) and consider the set

K = {2nVt : n < , t Xn} {}.

Note that K is a weakly compact subset of C(T (1)) which separates the pointsof T (1).

In the next application the coherent sequence 1 : ( < 1) ofnite-to-one maps needs to be turned into a coherent sequence of maps that areactually one-to-one. One way to achieve this is via the following formula:

1(, ) = 21(,) (2 |{ : 1(, ) = 1(, )}|+ 1).

Lemma 2.2.12.

(a) 1(, ) = 1(, ) for all < < < 1.(b) { : 1(, ) = 1(, )} is nite for all < < 1.Proof. Suppose < and that 1(, ) = 1(, ). Then 1(, ) =1(, ) = n and

|{ : 1(, ) = n}| = |{ : 1(, ) = n}|.

Since the set on the left-hand side is an initial segment of the set of the right-handside, the sets must in fact be equal. It follows that = . This proves (a).

To prove (b), by Lemma 2.2.2 the set

D = { : 1(, ) = 1(, )}18This is done by identifying a subset V of T 0(1) with its characteristic function V : T

0(1) 2.19Recall that a compactum X is Eberlein if its function space C(X) can be generated by a subsetwhich is compact in the weak topology.


is nite. Letm = max{(, ) : D , {, }}+ 1.

By Lemma 2.2.2, the proof of (b) is nished if we show that for < , if 1(, ) >m and 1(, ) > m, then 1(, ) = 1(, ). From the denition of m, we musthave that j = 1(, ) = 1(, ) > m, and that

{ : (, ) = i} = { : (, ) = i}.It follows that if j denotes the cardinality of this set, then

1(, ) = 2i(2j + 1) = 1(, ).

This completes the proof.

Dene 1 from 1 just as 1 was dened from 1; then the 1s are one-to-one and coherent. From 1 one also has a natural sequence r ( < 1) ofelements of dened as

r(n) = |{ : 1(, ) n}|.Note that r eventually strictly dominates r whenever < .

Denition 2.2.13. The sequences e = 1 ( < 1) and r ( < 1) can be usedin describing a functor

G G,which to every graph G on 1 associates another graph G on 1 as follows:

{, } G i {e1 (l), e1 (l)} G (2.2.9)

for all l < (r, r) for which these preimages are both dened and dierent20.

Lemma 2.2.14. Suppose that every uncountable family F of pairwise-disjoint nitesubsets of 1 contains two sets A and B such that AB G21. Then the same istrue about G provided the uncountable family F consists of nite cliques22 of G.Proof. Let F be a given uncountable family of pairwise-disjoint nite cliques ofG. We may assume that all members of F are of some xed size k.

Consider a countable limit ordinal > 0 and an A in F with all its elementsabove . Let n = n(A, ) be the minimal integer such that for all < in A{}:

(r, r) < n, 23 (2.2.10)

20As it will be clear from the proof of the following lemma, the functor G G can equallybe based on any other coherent sequence e : ( < 1) of one-to-one mappings and anyother sequence r ( < 1) of pairwise distinct reals.21Here, A B = {{, } : A, B, = }.22A clique of G is a subset C of 1 with the property that [C]2 G.23Recall that for s = t , we let (s, t) = min{k : s(k) = t(k)}.


e(, ) = e(, ) for some implies e(, ), e(, ) n. (2.2.11)Let

H(, A) = { 1 : e(, ) n(, A) for some from A {}}.Taking the transitive collapse H(, A) of H(, A), the sequence

e H(, A) ( A)

collapses to a k-sequence s(, A) of mappings with integer domains. Let r(, A)denote the k-sequence

r (n(, A) + 1) ( A)enumerated in increasing order. Hence every A F above generates a quadruple

(H(, A) , r(, A), s(, A), n(, A))

of parameters. Since the set of quadruples is countable, by the assumption on thegraph G we can nd two distinct members A and B of F above that generatethe same quadruple of parameters denoted by

(H, r, s, n).

Moreover, we choose A and B to satisfy the following isomorphism condition.

For every l n, i < k, if is the ith member of A, theith member of B and if e1 (l) and e

1 (l) are both dened,

then they are G-connected to the same elements of H .(2.2.12)

Let m be the minimal integer > n such that

(r, r) < m for all A and B. (2.2.13)

LetI = { : e(, ) m for some in A B}.

Let p be the k-sequence r m( A) enumerated in increasing order andsimilarly let q be the k-sequence r m( B) enumerated increasingly. Lett and u be the transitive collapse of e I ( A) and e I( B),respectively. By the Pressing Down Lemma there is an unbounded 1 andtuples

(H, r, s, n) and (I, p, q, t, u,m)

such that for all :

(H, r, s, n) = (H, r, s, n), (2.2.14)

(I , p, q, t, u,m) = (I, p, q, t, u,m). (2.2.15)


Moreover we assume the following analogue of (2.2.12) where k1 = |I \| for some(equivalently all) :

For every , and i < k1, if is the ith memberof I and if is the ith member of I in their increasingenumerations, then and are G-connected to the sameelements of the root I, and if happens to be the ithmember of A (or B) for some i < k, then must be theith member of A (B resp.) and vice versa.

(2.2.16)

By our assumption about the graph G there exist < in such that:

max(I) < , (2.2.17)

(I \ ) (I \ ) G. (2.2.18)The proof of Lemma 2.2.14 is nished once we show that

A B G. (2.2.19)Let i, j < k be given and let be the ith member of A and let be the jthmember of B.

Case 1: i = j. Pick an l < (r, r). By (2.2.10), l < n. Assume that e1 (l) ande1 (l) are both dened.

Subcase 1.1: e1 (l) < and e1 (l) < . By the rst choice of parameters, e

1 (l)

and e1 (l) are members of the set H which is an initial part of H(,A) andH(, B). Therefore, we have that the behavior of e and e on H is encoded bythe ith and jth term of the sequence s, respectively. In particular, we have

e1 (l) = e1 (l), (2.2.20)

where =the jth member of A . By our assumption that [A ]2 G we inferthat {, } G. Referring to the denition (2.2.9) we conclude that e1 (l) ande1 (l) must be G-connected if they are dierent.

Subcase 1.2: e1 (l) < and e1 (l) I \ . Let

= the jth member of B .

By the choice of parametrization, the position of e1 (l) in I is the same as theposition of e1 (l) in I so by (2.2.16) their relationship to the point e

1 (l) of the

root I is the same. Similarly, by the choice of the rst set of parameters, letting be the jth member of A , the position of e1 (l) in H(,B) and e

1 (l) in

H(,A) is the same, so from (2.2.12) we conclude that their relationship withe1 (l) is the same. However, we have checked in the previous subcase that e1 (l)and e1 (l) are G-connected in case they are dierent.


Subcase 1.3: e1 (l) I \ and e1 (l) < . This is essentially symmetric to theprevious subcase.

Subcase 1.4: e1 (l) I \ and e1 (l) I \ . The fact that {e1 (l), e1 (l)} Gin this case follows from (2.2.18).

Case 2: i = j. Consider an l < (r, r). Note that now we have l < m (see(2.2.13)).

Subcase 2.1: e1 (l) < and e1 (l) < . Then e

1 (l) and e

1 (l) are elements of

I which is an initial part of both I and I. Therefore the jth (= ith) term of uencodes both e I and e I (see the denition of above). It follows that

e1 (l) = e1 (l).

If l n then e1 (l) belongs to H and since the jth(= ith) term of s encodes bothe H and e H , it follows that

e1 (l) = e1 (l).

If l > n then e1 (l) and e1 (l) are not members of H so from (2.2.11) and the

denition ofH(,A) = H = H(,B)

we conclude that

e(e1 (l)) = e(e1 (l)) = l = e(e

1 (l)) = e(e

1 (l)).

Since e is one-to-one we conclude that

e1 (l) = e1 (l) = e

1 (l).

Subcase 2.2: e1 (l) < and e1 (l) I \ . Recall that is the ith (= jth)

member of B , so by the rst choice of parameters the relative position of e1 (l)in H(,B) must be the same as the relative position of e1 (l) in H(,A), i.e.,it must belong to the root H . Note that if j is the position of in I, then j

must also be the position of in I . It follows that the jth term of u encodesboth

e I and e I = e I.So it must be that e1 (l) belongs to the root I, a contradiction. So this subcasenever occurs.

Subcase 2.3: e1 (l) I \ and e1 (l) < . This is symmetric to the previoussubcase, so it also never occurs.

Subcase 2.4: e1 (l) I \ and e1 (l) I \ . Then {e1 (l), e1 (l)} G followsfrom (2.2.18).

This completes the proof of Lemma 2.2.14.


Lemma 2.2.15. If there is uncountable 1 such that []2 G, then 1 canbe decomposed into countably many sets such that []2 G.Proof. Fix an uncountable 1 such that []2 G. For a nite binary sequences of length equal to some l + 1, set

s = { < 1 : e(, ) = l for some in with s r}.

Then the sets s cover 1 and [s]2 G for all s. Remark 2.2.16. Let G be the comparability graph of some Souslin tree T . Thenfor every uncountable family F of pairwise-disjoint cliques of G (nite chains of T )there exist A = B in F such that AB is a clique of G (a chain of T ). However, it isnot hard to see that G fails to have this property (i.e., the conclusion of Lemma2.2.14). This shows that some assumption on the graph G in Lemma 2.2.14 isnecessary. There are indeed many graphs that satisfy the hypothesis of Lemma2.2.14. Many examples appear when one is trying to apply Martins axiom to someRamsey-theoretic problems. Note that the conclusion of Lemma 2.2.14 is simplysaying that the poset of all nite cliques of G is ccc, while its hypothesis is a bitstronger than the fact that the poset of all nite cliques of G is ccc in all of itsnite powers. Applying Lemma 2.2.15 to the case when G is the incomparabilitygraph of some Aronszajn tree, we see that the statement saying that all Aronszajntrees are special is a purely Ramsey-theoretic statement in the same way Souslinshypothesis is.

We nish this section by showing that the assumption (c) on a given C-sequence C ( < 1) on which we base our walks, and the function 1, is notsucient to give us the stronger conclusion that the tree T (1) can be decomposedinto countably many antichains (see Corollary 2.2.7). To describe an example, weneed the following notation, given that we have xed one C-sequence C ( < 1).LetC(0) = 0 and for 0 < n < , let C(n) denote the nth element ofC accordingto its increasing enumeration with the convention that C+1(n) = for all n > 0.We assume that the C-sequence is chosen so that for a limit ordinal > 0 and apositive integer n, the ordinal C+1(n) is at least n+1 steps away from the closestlimit ordinal below it. For a set D of countable ordinals, let D0 denote the setof successor ordinals in D. We also x, for each countable ordinal a one-to-onefunction e whose domain is the set 0 of all successor ordinals < and rangeincluded in , which cohere in the sense that e() = e() for all but nitelymany successor ordinals 0 0. For each r ([]


as above. This function will have the properties stated in Lemma 2.2.2, so thecorresponding tree

T (r1) = {(r1) : < 1}is a coherent tree of nite-to-one mappings that admits a strictly increasing func-tion into the real line. The following fact shows that T (r1) will typically not bedecomposable into countably many antichains.

Lemma 2.2.17. If r is a Cohen real, then T (r1) has no stationary antichain.

Proof. The family []


It follows that, given a < , the walk from to along the C-sequence Cx ( L(n, ). Similarly, we can ndMn2 Mn1Mn1 such that n2 = Mn21 > L(n1, n), and so on. Thiswill give us an -chain Mi (i n) of countable elementary submodels of (H2 ,)containing all relevant objects such that if we let n+1 = and i = Mi 1 fori n, then i > L(i+1, i+2) for all i < n. Pick an ordinal C0 above L(0, ).Then (see Lemma 2.1.6 above), i Tr(, ) for all i n and in fact 0 is thelast step of the walk from to . Let t = 0(, ), ti = 0(i, ) for i n. Thentn tn1 t0 t and t = t0|C0 |. Let

0 = { : t = 0(, ) and 0 = 0 }.Then 0, and since 0 M , we have that 0 M0. For 0, let

T = {x [1]|t| : x Tr(, ) for some }considered as a tree ordered by the relation . Call a subset of 0 large ifT contains a subtree T whose root is {} and in which a node x of cardinality< |t| must either have exactly one immediate successor or there is i n anduncountably many < 1 such that x {} T and 0(, ) = ti for all such that x {} Tr(, ). A simple elementarity argument using models M0 M1 Mn shows that 0 itself is large. Note also that if one decomposesa large subset of 0 into countably many subsets, at least one of them must belarge. So we can nd an ordinal 0 in the interval [, 0) and a large M0 suchthat 0 = 0(, ) for all . Let s = 0(0, ). Since the walk from to 0must pass through all the i (i n) we know that s t0 t1 tn. Choosea subtree T M0 of T witnessing that is large. Pick an immediate successor{, } of {} such that > 0. Then (see Lemma 2.1.6) for every ,

t0 = 0(, ) and Tr(0, ) = Tr(0, ) Tr(, ).Choose in M0 a branching node x1 {, } whose immediate successors x1{} T have the property that 0(, ) = t1 for all such that x {} Tr(, ).Let

l = osc(Tr(0, ) x1,Tr(0, )).Choose an arbitrary non-negative integer k < n. Working in M1, we choose animmediate successor x1 {1} T of x1 such that 1 > 1 Tr(0, ) and abranching node x2 x1 {1} of T corresponding to t2. Then working in M2, wechoose an immediate successor x2 {2} T of x2 such that 2 > 2 Tr(0, )and a branching node x3 x2 {2} of T corresponding to t3, and so on. Afterk steps we arrive at an immediate successor xk {k} T Mk of xk such thatk > kTr(0, ). Working in Mk we choose such that xk{k} Tr(, ).Let x = Tr(0, ), let y = Tr(0, ), and let z = Tr(0, ) x1. Then

x/E(x, y) = z/E(z, y) {x2 \ x1, x3 \ x2, . . . , x \ xk}.

2.3. Oscillations of traces 43

It follows that osc(, ) = l + k. Since k was an arbitrary non-negative integer n, this nishes the proof.

It should be clear that the idea of the previous proof with very minor mod-ications also works in proving the following multidimensional version of Lemma2.3.2.

Lemma 2.3.3. For every uncountable family G of pairwise-disjoint nite subsetsof 1 all of some xed size m, and every positive integer n, there exist integersli (i < m) and a, b G such that osc0(a(i), b(i)) = li + k for all i < m.25

Consider the following projection of the oscillation mapping osc0,

osc0(, ) = max{m : 2m|osc0(, )}.

One would need to choose a more involved projection in order to take full advan-tage of the multi-dimensional version of Lemma 2.3.2 but already this projectionis giving us interesting results.

Corollary 2.3.4. For every uncountable 1 and every positive integer k thereexist , such that osc0(, ) = k. Corollary 2.3.5. Every inner model M of set theory which correctly computes 1contains a partition of 1 into innitely many pairwise-disjoint subsets that arestationary in the universe V of all sets.

Proof. Our assumption about M means in particular that the C-sequence C ( : osc0(, ) = n} (n < )

belongs to M . So it suces to show that there must be < 1 such that forevery n < the set Sn is stationary in V . This is an immediate consequence ofCorollary 2.3.4.

Remark 2.3.6. In [67], Larson showed that Corollary 2.3.5 cannot be extended topartitions of 1 into uncountably many pairwise-disjoint stationary sets. He alsoshowed that under the Proper Forcing Axiom, for every mapping c : [1]2 1there is a stationary set S 1 such that for all < 1,

{c(, ) : S \ ( + 1)} = 1.

Note that by Corollary 2.3.4 this cannot be extended to mappings c with countableranges.

25Here a(i) denotes the ith member of a according to the increasing enumeration of a.


The original oscillation theory has been developed in the realm of functionswith domain and ranges included on , i.e., as a theory of the oscillation mapping

osc : ()2 + 1

that counts the number of times two such functions change in dominating eachother. Some of this will be reproduced in Section 9.4 below. In a quite similarmanner, one can also count oscillations between two nite partial functions x andy from 1 into as follows,

osc(x, y) = |{ dom(x) : x() y() but x(+) > y(+)}|,

where + denotes the minimal ordinal in dom(x) above if there is one.

Figure 2.4: The oscillation mapping osc1.

Denition 2.3.7. For < < 1, set (see Figure 2.4),

osc1(, ) = osc(1 L(, ), 1 L(, )).

2.3. Oscillations of traces 45

We have the following analogue of Lemma 2.3.2 which is now true even in itsrectangular form.

Lemma 2.3.8. For every pair and of uncountable subsets of 1 and everypositive integer n, there is an integer l such that for all k < n there exist and such that osc1(, ) = l + k.Proof. Fix a continuous -chainN of length 1 of countable elementary submodelsof (H2 ,) containing all the relevant objects and x also a countable elementarysubmodel M of (H2 ,) that contains N as an element. Let = M 1. Fix0 \ and 0 \ . Let L0 = L(, 0). By exchanging the steps in thefollowing construction, we may assume that 1(, 0) > 1(, 0) for = max(L0).Choose 0 C such that 0 > L0 and such that the three mappings 10 , 10 ,and 1 agree on the interval [0, ). By elementarity of M there will be +0 > ,+0 \ +0 , and +0 \ +0 realizing the same type as , 0, and over theparameters L0 and 0. Let L+1 = L(,

+0 ). Note that 0 min(L+1 ). It follows

that, in particular, the mappings 1+0 and 1+0 agree on L+1 . Choose an N N

belonging to M such that N 1 > L+1 and choose 1 C \N . Then we can nd1 > and +1 \ 1 and 1 \ 1 realizing the same type as , +0 , and +0over the objects accumulated so far, L0, 0, L+1 , N , and 1. Let L

++1 = L(, 1)

and let t+1 be the restriction of 1+1 on max(L+1 )+1. Then the set of whose

1 end-extends t+1 belongs to the submodel N and is uncountable, since clearlyit contains the ordinal +1 which does not belong to N . So by the elementarity ofN and Lemma 2.2.6 there is 1 \ such that 11 end-extends t+1 and

1(, 1) > 1(, 1) for all L++1 . (2.3.1)

Let t1 be the restriction of 11 on max(L++1 ). Then t1 M and every

whose 1 end-extends t1 satises (2.3.1) and

1(, ) = 1(, 1) for all L+1 . (2.3.2)

Note also that L(, 1) = L0 L+1 L++1 . Using similar reasoning we can ndt2 M T (1) end-extending t1, an 2 \ whose 12 end-extends t2,nite sets > max(L++2 ) > max(L

+2 ) > max(L

++1 ), and 2 \ such that

L(, 2) = L0 L+1 L++1 L+2 L++2 and such that the analogues of (2.3.1) and(2.3.2) hold for all whose 1 end-extends t2, and so on. This proceduregives us a sequence L0 < L+1 < L

++1 < < L+n < L++n of nite subsets of ,

an increasing sequence t1 t2 tn of nodes of T (0) M , and a sequencek (1 k n) such that for all 1 k n, and all whose 1 end-extends tk,

L(, k) = L0 L+1 L++1 L+k L++k , (2.3.3)

1(, ) > 1(, k) for all L++k , (2.3.4)1(, ) = 1(, k) for all L+k . (2.3.5)


Choose M whose 1 end-extends tn such that the set Ln = L(, ) liesabove L++n and such that all the 1k s agree on Ln. Note that by Lemma 2.1.6for each 1 k n,

L(, k) = L0 L+1 L++1 L+k L++k Ln. (2.3.6)Let l = osc(1 (L0 Ln), 1 (L0 Ln)) where is equal to one of theks. Note that l does not depend on which k we choose and that according to(2.3.4), (2.3.5) and (2.3.6), for each 1 k n, we have that osc1(, k) = l + k,as required.

Note that in the above proof the set can be replaced by any uncount-able family of pairwise-disjoint sets giving us the following slightly more generalconclusion.

Lemma 2.3.9. For every uncountable family G of pairwise-disjoint nite subsets of1 all of some xed size m, every uncountable subset of 1, and every positiveinteger n, there exist integers li (i < m) and a G such that for all k < n thereexist such that osc1(a(i), ) = li + k for all i < m.

Having in mind an application we dene a projection of osc1 as follows. Fora real x, let [x] denote the greatest integer not bigger than x. Choose a mapping : R such that (x) = n if x [x] In, where In = ( 1n+3 , 1n+2 ]. For x inR we use the notation x for (x). Choose a one-to-one sequence x ( < 1) ofelements of the rst half of the unit interval I = [0, 1] which together with thepoint 1 form a set that is linearly independent over the eld of rational numbers.Finally, set

osc1(, ) = (osc1(, ) x).Then Lemma 2.3.9 turns into the following statement about osc1.

Lemma 2.3.10. For every uncountable family G of pairwise-disjoint nite subsets of1 all of some xed size m, every uncountable subset of 1, and every mappingh : m , there exist a G and such that osc1(a(i), ) = h(i) for alli < m.

Proof. Let be half of the length of the shortest interval of the form Ih(i) (i < m).Note that for a G, the points 1, xa(0), . . . , xa(m1) are rationally independent,so a standard fact from number theory (see for example [46]; XXIII) gives us thatthere is an integer na such that for every y Im there exist k < na such that|k xa(i) yi| < (mod 1) for all i < m. Going to an uncountable subfamily of G,we may assume that there is n such that na = n for all a G. By Lemma 2.3.9there exists a G and li (i < m) such that for every k < n there is k > a suchthat osc1(a(i), k) = li + k for all i < m. For i < m, let yi = li xa(i) and let zi bethe middle-point of the interval Ih(i). Find k < n such that for all i < m,

|k xa(i) (zi yi)| < (mod 1).Then osc1(a(i), k) = h(i) for all i < m, as required.

2.4. The number of steps and the last step functions 47

Corollary 2.3.11. There is a regular hereditarily Lindelof space which is not sepa-rable.

Proof. For < 1, let f {0, 1}1 be dened by letting f() = 1, f() = 0 for > , and f() = min{1, osc1(, )} for < . Consider F = {f : < 1} asa subspace of {0, 1}1. Clearly F is not separable. That F is hereditarily Lindeloffollows easily from Lemma 2.3.10. Remark 2.3.12. The projection osc0 of the oscillation mapping osc0 appears in[109] as the historically rst such map with more than four colors that takes all ofits values on every symmetric square of an uncountable subset of 1. The variationsosc1 and osc1 are on the other hand very recent and are due to J.T. Moore [78]who made them in order to obtain the conclusion of Corollary 2.3.11. ConcerningCorollary 2.3.11 we note that the dual implication behaves quite dierently since,assuming the Proper Forcing Axiom all, hereditarily separable regular spaces arehereditarily Lindelof (see [111]).

2.4 The number of steps and the last step functions

In this section we show that a very natural characteristic associated to the mini-mal walks between countable ordinals lead to functions that have coherence andnontriviality properties very much reminiscent of the Hausdor gap phenomenonthat will be a subject of our study in Section 3.1 below.

Denition 2.4.1. The number of steps of the minimal walk is the two-place function2 : [1]2 dened recursively by

2(, ) = 2(,min(C \ )) + 1,with the boundary condition 2(, ) = 0 for all .

This is an interesting mapping which is particularly useful on higher cardinal-ities and especially in situations where the more informative mappings 0, 1 and lack their usual coherence properties. Later on we shall devote a whole sectionto 2 but here we list only few of its basic properties. We start with the coherenceproperty that this function enjoys.

Lemma 2.4.2. sup{|2(, ) 2(, )| : < } k. We mayassume that the sequence of ks is strictly increasing and let = supk k. Then is a limit ordinal , so the lower traces of walks from to and to have acommon upper bound < . Then by Lemma 2.1.6, for every ordinal [, ),we have that

0(, ) = 0(, )0(, ) and 0(, ) = 0(, )0(, ). (2.4.1)


It follows that for every [, ),

|2(, ) 2(, )| |2(, ) 2(, )|, (2.4.2)

and so in particular, k / [, ) for all k such that k > |2(, ) 2(, )|, acontradiction.

We mention also the following unboundedness property of this function whichintroduces another theme to be explored fully in later sections of this book.

Lemma 2.4.3. For every uncountable family A of pairwise-disjoint nite subsetsof 1, all of some xed size n, and for every integer k, there exist an uncountablesubfamily B of A such that for all a < b in B, we have 2(a(i), b(j)) k for alli, j < n.26

Proof. The proof is by induction on k. So suppose that our given family A alreadysatises that 2(a(i), b(j)) k for all i, j < n and all a < b in A. We shall nduncountable B A such that 2(a(i), b(j)) k + 1 for all i, j

tordorcevic - walks on ordinals

Documents

squarebracket operation

square sequences7

special square sequence

general walks

metric theory of ordinals

countable ordinals5

countable ordinals2

number of steps