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Partition propertiees and Halpern-Lauchi Theoren on the Cmin forcing

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Partition propertiees and Halpern-Lauchi Theoren on the Cmin forcing
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Full Text











PARTITION PROPERTIES AND HALPERN-LAUCHLI THEOREM ON THE
C,mi FORCING
















By

YUAN-CHYUAN SHEUf


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005















ACKNOWLEDGEMENTS


I would like to thank my advisor, Jindrich Zapletal, for his patience and almost

infinite discussion with me on the problems I was trying to solve. Without him, I

simply could not have got this far.

I would also like to thank the other members of my supervisory committee:

Dr. Jean A. Larson, Dr. William Mitchell, Dr. Rick L. Smith, and Dr. Greg Ray. I

have learned a lot from them either through taking classes taught by them, personal

conversations, or hearing them speak in seminars and conferences.

I am very grateful to the Department of Mathematics for supporting me with

a teaching assistantship during my years as a graduate student. I also would like

to thank the College of Liberal Arts and Science for supporting me with a Keene

Dissertation Fellowship in the spring 2005 term.

Finally, I must acknowledge the contribution of my family and all my friends

in the Department of Mathematics for their caring and support.



















TABLE OF CONTENTS


ACKNOWLEDGEMENTS .........

ABSTRACT .................

CHAPTERS

1 INTRODUCTION ..........


2 PRELIMINARIES ..........

2.1 Basic Definitions ........
2.2 The Property of Baire .....
2.3 The space 2 ..........
2.4 The Cmin Trees.........
2.5 Determinacy ..........

3 BLASS PARTITION THEOREM .

3.1 Blass Theorem on the Perfect S
3.2 Partition Theorem on the Cmin

4 CANONICAL PARTITION THEOR

4.1 Perfect Tree Case .......
4.2 Cmin Tree Case .........

5 HALPERN-LAUCHLI THEOREM

5.1 Introduction ..........
5.2 C,in Tree Case .........

6 OTHER FORCING NOTIONS ...

6.1 The E0 Trees ..........
6.2 The Silver Forcing .......
6.3 The Packing Measures .....

REFERENCES ................

BIOGRAPHICAL SKETCH .........


. . ii

. . iv







. 3

. . 3
. .. 3
. . 5

. . 6
S. .. .
. .. 7


ets .......8..... .... 8

Tress ... ... .... ... 10

EM .. .. .. 17

. .. 17
. . 18

. . 24
. 24
24

. 25

. 33

. 33
. 34
.. . 36

. 38

. 39















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

PARTITION PROPERTIES AND HALPERN-LAUCHLI THEOREM ON THE
Cmn FORCING

By

Yuan-Chyuan Sheu

May 2005

Chairman: Jindrich Zapletal
Major Department: Mathematics

In this dissertation, we focus on a special breed of forcing notion called Cmin

forcing, which is forcing equivalent to the form B(R) \ I for some Cmin-ideal I.

We define a special perfect subsets of the space 2", called Cmin trees. We study

the partition properties and Halpern-Liuchli theorems on these trees. We obtained

three principal theorems which will be our discussion Chapters 3,4 and 5 respectively.

We also show that the some other forcing notions do not have similar partition

properties. This is in Chapter 6.















CHAPTER 1
INTRODUCTION

This dissertation is based upon the deep combinatorial theorem of Halpern-

Liuchli [6] on the perfect trees which Halpern and Liuchli developed in the 1960's.

The Halpern-Luchli Theorem is obtained as a byproduct of the proof of Con(ZF +

-,AC + BP), where BP is the statement that there exists a prime ideal in every

Boolean algebra, a consequence of the Axiom of Choice equivalent to Compactness

Theorem for the First-Order Predicate Calculus. It has been noticed since then that

the lemma of the proof might be of independent interest.

One of the main applications of the Halpern-Liuchli Theorem is the proof of

the well known Blass Theorem [1] on perfect trees, which states

T -- [S]

Let a Cmin tree [14] be a nonempty binary tree T C 2<" such that every node

t E T can be extended both into a splitnode of even length and into a splitnode

of odd length. The study of the Cmin trees was originated from Geschke et al. [4],

where the authors defined the Cmin function (see Chapter 2 for definition). The Cmin

a-ideal I generated by the C,i, homogeneous sets is nontrivial and the subsets of 2"

that contain all the branches of a Cmin tree are the typical I-positive sets. The Cin

trees as a forcing notion ordered by inclusion is proper.

We have shown that for the Cmin trees we have a similar but different

T -+ [Si],2,-1,(_-1)!

This is contained in Chapter 3.









In 1930 Ramsey [11] proved the famous pigeon-hole principle for finite sets. A

new situation arises if partitions into an arbitrary number of classes are considered.

For this case, Erd6s and Rado [3] proved the so-called canonical version of Ramsey

Theorem.

Based on the Cmin version of Blass Theorem, we also proved the canonical

theorem on Borel equivalence relations on Cin trees. This means that, fixing a

shape and parity pattern, there exists a finite set G of equivalence relations on [2"]"

such that for any Borel equivalence relation f on [2W"] there is an equivalence relation

g E G and a Cmin tree T C 2" such that f t [T]" = g [ [T]". This is in Chapter 4.
We obtained a Cmin tree version of the Halpern-Liuchli Theorem. In a 1984

paper, Richard Laver [8] generalized the existing Halpern-Liiuchli Theorem to the
infinite product of perfect trees. From there, equivalent results on Hilbert cubes and

selective ultrafilters can be derived. We have developed the Halpern-Liiuchli Theorem

on the infinite product of Cmin trees and we have the equivalent results on Hilbert

cubes and selective ultrafilters as well. This is contained in Chapter 5.

We have also shown that the Cmin trees are special by constructing counterex-

amples that demonstrate that no such results are possible for a variety of other kinds

of forcing notions such as Eo Forcing, Silver Forcing and the forcing notion associated

with the Packing Measure. This is the main content of Chapter 6.














CHAPTER 2
PRELIMINARIES

2.1 Basic Definitions

A relation is a set R all of whose elements are ordered pairs. f is a function
iff f is a relation and

Vx E dom(f) 3!y E ran(f) ((x,y) E f).

f : A B means f is a function, A = dom(f), and ran(f) C B. If f : A -+ B

and x E R, the set of all real numbers, f(x) is the unique y such that (x, y) E f;
ifC C A, f r C = f n C x B is the restriction of f to C, and f"C = ran(f [ C)

={ (f() : x e C}.
A total ordering is a pair (A, R) such that R totally orders A; that is, A is a

set, R is a relation, R is transitive on A:

Vx, y, z E A (xRy and yRz xRz),

trichotomy holds:
Vx, y E A(x = y or xRy or yRx),

and R is irreflexive:

Vx E A it is not the case that xRx.

As usual, we write xRy for (x, y) E R.
We say that R well-orders A, or (A, R) is a well-ordering iff (A, R) is a total
ordering and every non-empty subset of A has an R-least element.

Definition 2.1.1. (a) A partial order is a pair (P, <)such that P I 0 and < is a
relation on P which is transitive and reflexive (Vp E P(p < p)). p < q is read p
extends q". Elements of P are called conditions.









(b) (P, <) is a partial order in the strict sense iff it in addition satisfies

Vp, q(p < q and q < p -+ p = q). In that case, define p < q iff p q and p # q.

Definition 2.1.2. An equivalence relation on a set X is a binary relation ~, which

is reflexive, symmetric, and transitive: For all x,y, z E X, we have that x ~ x,

x y -+ y x, if x ~ y andy z then x z.

Definition 2.1.3. A tree is a partial order in the strict sense (T, <), such that for

each x E T, {y T : y < x} is well-ordered by <.

The set of all finite sequence of O's and l's, 2<1 is a tree with any s, t E 2',

s < t iff s C t. For any T C 2< let [T] be the set of all branches through the tree

T. Any TC 2< is a tree ifVt T and s = t [ n for some n, then sET.

Definition 2.1.4. A skew tree is a tree so that on each level there is at most one

branching node.

For distinct a, p E [T], let d(a, 3) be the level of the highest common node

of the paths a and i through T. Here we assume for any n-set it is the case that

the elements are listed in lexicographic order, and we may assume that the trees are

skew.

Definition 2.1.5. By the shape of an n-element set {ao, .., an-1} C [T], we mean

the linear ordering of {1, ..., n 1} given by i -< j -+ d(ai-1, ai) is in a lower level

than d(aoj_, aj).

Definition 2.1.6. By the parity pattern p, of an n-element set {ao, ..., an-1} E [T]n,
we mean p E 2"-' given by p(i) = 0 if the i-th (counting from 0) lowest splitnode of
the n-set is on the even level.

Definition 2.1.7. Let T = (Ti : i < d) be a sequence of trees. Define (&T= i

to be the set of all n-tuples X from the product of {Ti : i < d} such that IX(i)I =









IX(j)| for any i,j < d; for A C w, let (A T be the set of all n-tuples Y from the
product of {Ti : i < d} such that IY(i)l = IY(j)l E A for any i,j < d.

Definition 2.1.8. An algebra of sets is a collection S of subsets of a given set S
such that

(i) S S,
(ii) if X S and YES then X U Y E S,
(iii) if X E S then S \X E S.
A a-algebra is additionally closed under countable unions:
(iv) IfXn E S for all n, then U'oXn E S.
For any collection X of subsets of S there is a smallest a-algebra S such that

S D X; namely the intersection of all a-algebras S of subsets of S for which X C S.

Definition 2.1.9. A set of reals B is Borel if it belongs to the smallest a-algebra B
of sets of reals that contains all open sets.

2.2 The Property of Baire

Let us call a set A C 2" nowhere dense if the complement of A contains a
dense open set. Note that A is nowhere dense just in case that for every non empty
open set G, there is a nonempty open set H C G such that A n H = 0. A set A is
nowhere dense if and only if its closure A is nowhere dense.
A set A C 2" is meager if A is the union of countably many nowhere dense
sets.

Definition 2.2.1. A set A has the Baire property if there exists an open set G such
that AAG = (A \ G) U (G \ A) is meager.

Clearly, every meager set has the Baire property. Note that if G is open, then
G \ G is nowhere dense. Hence if AAG is meager then (2W \ A)A(2W \ G) = AAG is
meager, and it follows that the complement of a set with the Baire property also has









the Baire property. It is also easy to see that the union of countably many sets with

the Baire property has the Baire property and we have:

Lemma 2.2.2. The sets having the Baire property form a a-algebra; hence every

Borel set has the Baire property.

2.3 The space 2W

Let w be the set of all natural numbers. The space 2W is the set of all infinite

sequences of numbers 0 or 1, (an : n E w), with the following topology: For every

finite sequence s = (ak : k < n), let

O(s) = {f 2 : s C f} = {(ck : k E w) : (Vk
The sets O(s) form a basis for the topology of 2". Note that each O(s) is also closed.

The space 2" is separable and is metrizable: consider the metric
1
d(f, g)= 2-

where n is the least number such that f(n) : g(n). The countable set of all eventually

constant sequences is dense in 2". This separable metric space is complete, as every

Cauchy sequence converges.

2.4 The C,,, Trees

Definition 2.4.1. Let Cmin : [2"]2 -+ 2 be the mapping defined by

Cmin(x, y) = A(x, y) mod 2,

where A(x, y) is the least number n such that x(n) # y(n), and let I, the Cmin-ideal

be the a-ideal a-generated by the Cmin homogeneous sets.

The above definition can be found in Geschke et al. [4]. It is not difficult to

verify that Cmin homogeneous sets must be meager. It is so because if A is Cmin-

homogeneous, say Vx, y E A Cmin(x, y) = 0, then all the split nodes of A are on even
levels.










Now, it is clear that A is meager. And hence the ideal I is non-trivial.

Definition 2.4.2. A Cmin tree is a nonempty tree T C 2
t E T can be extended both into a splitnode of even length and into a splitnode of odd

length.

The following theorem can be found in Shelah and Zapletal [12].

Theorem 2.4.3. Every analytic subset of 2W is either in the ideal I or it contains

all branches of some Cmin tree. Under AD (see below) this extends to all subsets of

2".

2.5 Determinacy

With each subset A of 2" we associate the following game GA, played by two

players I and II. First I chooses a number ao E {0, 1}, then II chooses a number

b0 E {0, 1}, then I chooses al, then II chooses bl and so on. The game ends after

w steps; if the resulting sequence (ao, bl, al, b, ...) is in A, then I wins; otherwise II

wins.

A strategy (for I or II) is a rule that tells the player what move to make

depending on the previous moves of both players. A strategy is a winning strategy

if the player who follows it always wins. The game GA is determined if one of the

players has a winning strategy.

The Axiom of Determinacy (AD) states that for every A C 2W, the game GA

is determined.















CHAPTER 3
BLASS PARTITION THEOREM

3.1 Blass Theorem on the Perfect Sets

Let 2W be ordered lexicographically. For X C 2", let [X]n be the set of n-

element subsets of X. When we describe a finite subset of 2" by listing its elements,

we always assume that they are listed in increasing order. Thus [2"]n is identified

with a subset of the product space (2W"), from which it inherits its topology. A subset
A of 2W is perfect if it is nonempty and closed and has no isolated points. This is

equivalent to saying that A = [T], where T is a tree such that for every t E T there

exist so, sl D t, both in T, that are incomparable, i.e., neither so D si nor sl D so.
A. Blass [1] proved the following

Theorem 3.1.1. Let P be a perfect subset of 2W and let [P]n be partitioned into

a finite number of open pieces. Then there is a perfect set Q C P such that [Q]n
intersects at most (n 1)! of the pieces.

It was pointed out in Blass [1], that the theorem remains true if 2W is replaced

by the real line R with its usual topology and order. To see this, it suffices to

observe that every perfect subset of R has a subset homeomorphic to 2" via an order-
preserving map and that any one-to-one continuous image in R of a perfect subset

of 2" is perfect in R Second, the hypothesis that the pieces of the partition are
open can be greatly relaxed. Mycielski [9, 10] has shown that any meager set or
any set of measure zero in [R]n is disjoint from [P]n for some perfect P C R For
the meager case, he obtains the same result with R replaced by any complete metric

space X without isolated points. It follows that, if [R]" is partitioned into finitely
many pieces that have the Baire property, then their intersections with [P]" are open









for some perfect P. Similarly, if the pieces are Lebesgue measurable, they become G6
sets when restricted to [P']" for suitable perfect tree P'; since G6 sets have the Baire
property, we can apply the preceding sentence, with P' as X, to get a perfect P C P'
such that the pieces intersected with [P]n are open in [P]n. Thus the theorem, as
extended by the first remark as above, implies the following

Corollary 3.1.2. If [R]" is partitioned into finitely many pieces that all have the

Baire property or are all measurable, then there is a perfect set Q C R such that [Q]n
meets at most (n 1)! of the pieces.

Some hypothesis about the pieces is necessary, however, for Galvin and Shelah
[5] have shown that there is a partition of 2' into infinitely many pieces such that,
for any Q C 2W of the cardinality of the continuum, [Q]2 intersects all the pieces.
As mentioned in Blass [1], we may assume that the all the perfect trees are

skew (Definition 2.1.4.). Since -< (see Definition 2.1.5.) tells us in what order the
paths ai split apart as we proceed up the tree and there are (n 1)! linear orderings

-< of {1, ..., n 1} and all of them are obviously realized as shapes within any perfect
subset of [T], we see that the (n 1)! in the theorem is optimal.
We shall need a few definitions: Let T= (To,..., Tr-1) be a skew r-tuple of

perfect trees, which means not just each tree Ti is skew but also that no two distinct
T,'s have splitting nodes at the same level. Let n= (no, ..., nr-) be a r-tuple of
positive integers with sum n. By an n-set in T we mean an r-tuple a whose ith

entry oa is an ni-element subset of (Ti). The shape -< of such an n-set a is the linear
ordering of the pairs (i,j), with 0 < i < r and 1 < j < n,, given by the levels of
d(aj_1, ai,j), where ai, is the j-th element of cr in lexicographic order. The above
theorem is a special case when r = 1 of the more generalized theorem.

Theorem 3.1.3. (Polarized Theorem) Let r, T, n, n be as above and let -< be any
shape of n-sets. Let the collection of all n-sets in T be partitioned into finitely many









open pieces. Then there exist perfect trees T' C Ti such that all n-sets in T' with

shape -< lie in the same piece of the partition.

3.2 Partition Theorem on the C,i. Tress

Lemma 3.2.1. Every Cmin-tree has a Cmin skew subtree.

Proof: Given any Cmin-tree T. To obtain a skew subtree, we must make sure

that at each level, there is at most one branching node. Also, we have to make sure

that for each node t E T, there are branching extensions to, tl E T of t with Ito| even

and Itl odd. We will construct the Cmin skew subtree T' of T by induction on the

number of branching nodes below the nodes we are currently looking at, each stage

make sure the above two requirements are satisfied:

Suppose we have done up to stage n. At stage n + 1, look at all the maximal

(meaning there is no node extends it so far in the construction) nodes constructed so

far and make sure each of them has two extensions of branching nodes, one of them

has even length, one of them has odd length, and so that at each level there is at

most one branching node. Now it is easy to see that T' is a Cmin subtree of T.


The parity pattern p for an n-set r in a sequence of trees T is defined by

p (i) = 0 iff the i-th lowest splitnode in aj is on the even level.

Theorem 3.2.2. (Polarized Theorem for the Cmin trees) Let T=< To, T1,...,Tr-1 >

be a set of Cm,, trees, n=< no,...,n,-1 >, and let -< be any shape of n-sets, p be

a parity pattern of n-sets. Let the collection of all n-sets in T be partitioned into

finitely many open pieces. Then there exist Cmin trees Ti C Ti such that all n-sets in
T' with shape -< and parity pattern p lie in the same piece of the partition.

It easily follows from the theorem as a special case when r = 1, we also have

the following corollary.









Corollary 3.2.3. Let T be a Ci-tree, and let [T]" be partitioned into a finite number

of open( in [T]n) pieces. Then there is a Cmin-tree S C T such that [S]n intersects at

most 2"-' x (n 1)! of the pieces.

Note that the 2n-1 x (n 1)! in the conclusion of the corollary is optimal since

there are (n 1)! many shapes and 2"-' many parity patterns for the sets B E [T]n,

where T is a Cmin tree.

Before we prove the theorem, let me point out that it is easy to show (using

a typical fusion argument; see the theorem below) that any meager set in [2"]n is

disjoint from some [P]n for some Cmin-tree P. Hence we can generalize the theorem

so that the partition is Baire. The proof follows the original proof of Blass on perfect

sets.

Theorem 3.2.4. Given a Cmin-tree T, a natural number m and a meager set A C

[T]m, there exists a Cmin subtree P of T, such that [P]m n A = 0.

Proof: Before we prove the theorem, we need some definitions. The proof

uses the technique of fusion. Let p be a Cnin-tree. A node s E p is a splitting node if

both s^O and s^l are both in p; a splitting node s is an n-th splitting node if there

are exactly n splitting nodes t such that t C s. For each n > 1, let p <: q if and only

if p < q and every n-th splitting node of q is an n-th splitting node of p.

A fusion sequence is a sequence of conditions {p : n E w} such that pn
Pn+-. It is clear that if {p, : n E w} is a fusion sequence then nnEwPn is a Cm,i-tree.
If s is a node in p, let p [ s denote the tree {t p : t C s or t D s}. If B

is a set of incompatible nodes of p and for each s E B, q, is a Cmin-tree such that

q, C p r s then the amalgamation of {q, : s E B} into p is the tree


{t E p : if t D s for some s E B then t E q,}.









Now we start the proof of the theorem by constructing a fusion sequence { : n E w}

of Cmin-trees by induction on the level of the split nodes n: We may assume that

A = UiEAi, where each Ai is closed nowhere dense.
Let d be large enough such that 2d > m. We will begin our induction stage 0 on

the set of d-th branching nodes {so,..., S2d-}1) f T. For each si find two nodes s~O^'t

and s 1^t! such that each |tI is even and Itl| is odd and for any n-tuple (xo,..., xm-1)

in [T]m with each xj in a different Cmin tree T [ s-a^-t, where a, b {0, 1}, we have

that (xo, ..., Xm-1) Ao. This can be done using the fact that Ao is closed nowhere

dense. Let Po be the amalgamation of {T [ s a^-t : a, b = 0,1}. It is clear that Po is

Cmin. Suppose now we have obtained Po, ..., Pn at stage n. At stage n + 1 repeat the

process on the d + n + 1-th branching nodes of Pn such that all the m-tuples selected

as above from different Cmin subtrees of Pn avoid the set {Ao,..., An+i} and obtain

Pn+1.
Let P = n,,,P,. Now we have that [P]" n A = 0.


Proof of theorem 3.2.2.: By remove node s from T," we mean that T is

to be replaced by its largest subtree not containing s; kill the branching node s of

T" means to remove( in the sense just explained) one of the two immediate successors

of s. The choice of which successor to remove is arbitrary except when a specific node

t above s is to be retained"; then the immediate successor of s that is not below t
is to be removed.

We proceed by downwards induction on r, with n fixed. Since each ni is

required to be positive, the highest possible value of r is n, and if r = n each ni = 1.
An n-set from T is then just an n-tuple a of paths ai through Ti. Fix such an

n-set a. Since the partition is open, all n-sets sufficiently close to a lie in the same

partition class. Thus, for sufficiently long finite initial segments si of as, the trees
Ti = Ti r si satisfy the conclusion of the theorem.









We turn our attention to the nontrivial case r < n and assume that the
polarized theorem for r + 1. Without loss, we may assume that for any n-set o the
lowest-level branching point f(a) is in To. In each of the remaining trees Ti(1 < i <
r), all of the paths aij pass through the same node s, at the level of f(o). We call
the r-tuple (f(ar), sl,..., sr-i) the signature of the n-set a.

Lemma 3.2.5. There are Cmin Ti* C Ti so that any shape -< and parity pattern p,
all the n-sets having the same signature lie in the same partition class.

Proof: Each tree Ti* will be obtained as the intersection of an inductively

defined decreasing sequence of Cmin subtrees of Ti. At each stage n, build Cmin
subtrees Ti" of Ti and select some number ,n so that any s E 2<1 that is in the tree
T77 and below level In will be in the trees Tim for all m > n. In the end T* = nn<,T"n.
To begin the induction, set To = Ti, l=0.
Suppose that, at a later stage n, we have obtained Cmi, subtree Tin and some

number ,I < w. Choose a level 1, so high that in each Ti(i # 0) every node s at level
In has at least two successors tl, t2 that are branching nodes and such that It | is odd
and It21 is even; this can be done since each Ti" is Cmin. Kill all the branching nodes
of Ton between level In and 1, inclusive and let 1,,n = In.
Next, choose a branching node f of To" above level l~ say at 1~ and kill all the
branching nodes between l, and l 1 inclusive, retaining f, and let ln+i = lN.
We now seek to make sure that all n-sets with shape -< and parity pattern p

with f as the first branching node have their partition classes determined by their

signatures. Enumerate all the possible signatures (f,sl, ...,sk) that begin with f;
there are only finitely many, say m, of them. Consider each such signature in turn.
Construct Cmin subtrees T, ...,T"'m of each Ti by a finite induction as follows:
Suppose we have constructed up to some TI'j at some stage j, for stage j + 1,
consider:









The n-set r in T"'J with shape -< and parity pattern p and the j + 1-th

signature (f, sl, ..., Sr-) yields, by splitting the first component, an n*-set

c* = ({ao,..., oq-1, o,q, ,---, O,no-l a1, ', r-1),

which is in

Tn"j, = (To'j [ fr O, T~"j f'l,T'n, r si,...,TrTJ r sr-1),

where n* is the (r+ 1)-tuple (q, no q, n, ..., nr-i). The shape -<* and parity pattern
p* of ar* is uniquely determined by -< and p, and r* determines o. Partition the

n*-sets in T"n,'* with shape -<* and parity pattern p* by putting two such o*'s in the

same piece if and only if the corresponding o's are in the same piece of the original
partition. Since a is a continuous function of or*, this is an open partition of n*-sets,
and we can apply the induction hypothesis to find Cmin subtrees of the Tin7'*'s such

that all their n*-sets with shape -<* and parity pattern p* lie in the same piece of
partition class. Prune the trees Ti', correspondingly and call them Ti'+' so that the

new
Tnj+l [ f^O, To'n+1 f- 1, Tn,7j+l r sl, s ., Tn+.. 1 [ S-1

are the subtrees of the T "~'*'s given by the induction hypothesis.
Let T+1 = nj Now, let T* = n
subtree. By carefully selecting the f's, To can be arranged to be C,in as well. We

may assume this T* is good for each fixed shape and parity pattern by repeating the
process finitely many times. This completes the proof.


We now seek to eliminate the dependence of the partition class on the signature

for the n-sets with fixed shape -< and parity pattern p, by reducing the trees still
further.









Without loss, let us assume that po(0) = 0; i.e., the lowest split node of oo is
on the even level. To each r-tuple s with si e Ti* we associate a signature s of T*
as follows. Let gs be the highest even level branching node C so in T* (It will not
matter how we define if so is below the lowest even level branching node of To.)

For i 0 0, let si be the predecessor or the lexicographically first successor of si at the
level of go. Clearly, s = (so, ...-r 1) is a signature of T* for the n-sets with shape -<
and parity p.
We apply the Halpern-Liuchli partition theorem [6] to the partition of T* x

... x Tr- = T* sending two r-tuples to the same partition class determined by the

last lemma with shape -<, parity pattern p if their signatures are in the same partition
class. It asserts that there is a natural number h such that one of the partition classes,
say c, contains (h, k)-matrices for all natural numbers k. This means that, for each
k, there is a sequence x of nodes xi at level h in T7* and there is a sequence A of
subsets Ai C T* such that
(1) every successor of xi at level h + k in Ti* is below a node in Ai, and
(2) all r-tuples s E f A lie in the partition class c.
Fix such an h. There are only finitely many tuples x as above, so the same
x works for arbitrarily large k; hence we may assume it is for all k. Fix such an x.
The process of pruning the trees T* begins with setting Si = T* [ xz. To simplify
notation, we assume that h = 0; the general case involves adding h to every level
mentioned in the sequel.
Now proceed by induction, at each stage n build Ci, subtrees T*'"n of T*
and select some number I, so that any s E 2 level l, will be in the T7'" for all the m > n. In the end T[ = nn<,Ti'". To begin
the induction, set T'i* = Ti* and lo = 0.
Suppose that at a later stage n, we have obtained Cmin subtrees Ti*'n and some
number In < w. In stage n + 1, choose 1, so large that, in each T*'"(i $ 0), every









node s at level In has at least two branching nodes to, tl above s and below level I'n
so that Ito| is even and Iti| is odd. Next, choose an even level branching node f in

To'' above level 1 say at level 1~ > l'. By Halpern-Liiuchli, find Ai C Ti so that
(1) Ai dominates all nodes of level I" in T*'", and
(2) HAC c.

By (1), find so E Ao above f. Then s0 is above or equal to f, so its level 1* is
greater or equal to l'. Kill all the even level branching nodes of To*n from level In to
level 1* 1 inclusive retaining so. As before, this ensures that go is the unique even

level branching node of To*' between levels In and 1 inclusive. For any i : 0 and
any si E Ai, let be, as before, the predecessor or lexicographically first successor
of si at level 1*. Then Bi = {ilsis E Ai} dominates level l' of T*'" because 1* > l'
and (1) holds. Furthermore, by (2) and the definition of the partition for r-tuples

other than signatures, {go} x fi40 Bi is included in class c of the original partition
of signatures. For i : 0, remove all nodes of Ti*'" that are not comparable with any
node in Bi, and let the resulting tree be T*'"+1 and let ln+1 = 1* + 1. All nodes at
level 1 in T,n+l(i 0) belong to Bi, so all signatures that start with s0 are in class
C.
Now let T.' = n,< ,Tj". It is clear that for each i $ 0, we have that T[ is Cmin.
To can be arranged to contain a Cmin tree as well by carefully choosing the f's.















CHAPTER 4
CANONICAL PARTITION THEOREM

4.1 Perfect Tree Case

In 1930 Ramsey [11] proved his famous pigeon-hole principle for finite sets:

Theorem 4.1.1. Let n, r be positive integers and let X be a countable infinite set.

Then for every partition A : [X]n --+ {0,...,r 1} of the n-element subsets of X

into r many classes, there exists an infinite subset Y C X such that the restriction

A f [y]n is a constant mapping.

A new situation arises if partitions into an arbitrary number of classes are

considered. For this case, Erd6s and Rado [3] proved the so-called canonical version

of Ramsey theorem:

Theorem 4.1.2. Let n be a positive integer and let X be a countable infinite set,

which is totally ordered. Then for every mapping A : [X]n -+ X there exists an
infinite subset Y C X and a subset I C {0, ..., n 1} such that for all {ao, ..., an-},

{bo,..., bn,_} E [Y]" with ao < ... < an-~ and bo < ... < bn- we have that

A({ao,..., an1}) = A({bo,..., bn-1}) if {ai : i E I} = {b : i E I}.

For many other structures like for example arithmetic progressions, parameter-

words and finite vector spaces, canonical partition results are known.
H. Lefmann [7] considered perfect-tree version of these theorems in the space
2" with the metric

1
d(x, y)= -iff k = min{n: x(n) # y(n)}.

and have the following result:









For any A E [T] A = {ao,..., a,-1} and J C {1,...,n- 1}, let D(A) : J =

{d(a-1, a,) : j E J}, where d(x, y) is the distance between x, y.

Theorem 4.1.3. Let n be a positive integer and let ({1,...,n 1},-<) be a total
order. Further let T C 21 be a perfect subset and let X be a metric space. Then
for every Baire-mapping A : [7T1] -- X there exists a perfect subset P C T and
subsets I C {0,..., n 1} and J C {1,...,n 1} such that for every A,B E [7P]n with
A = {ao,..., aLn-1}<,ex and B = {(o, .., /n-1} A(A) = A(B) iff {a : i I} = { : i E I} and {d(aj-_,aj) : j J} =
{d(1pj-, [j) : j J}.

4.2 C,,a Tree Case

Lemma 4.2.1. For any Borel equivalence relation ~ on a Cmi, tree T C 2", there
is a Cmin tree P C T so that either x ~ y for all x, y E P or x ~ y iff x = y for all
x, yE P.

Proof:
Let ~ be a Borel relation on T. This induces a mapping A* : [T]2 {0, 1} by
A*({ao, al}) = 0 iff ao ~ a,. Clearly, A* is a Borel mapping. By the Cmin version
of Blass Theorem, there exists a Cmin subtree P C T, either A*([P]2) = i for some
i = 0, 1, or A*(ao, a1) = 0 iff ao, al split on even (odd) level. We have the following
four cases:
(1). If this P is such that A*([P]2) = 0, then obviously we must have that
a0o ~ a for any ao, a1 E P.
(2). It is the case that A*([P]2) = 1, then we have that ao ~ a1 iff ao = a,.
(3). It is the case that A*(ao, al) = 0 iff ao, al split at an even level. This
does not happen, since is a Borel equivalence relation.
(4). It is the case that A*(ao, ai) = 0 iff ao, a1 split at an odd level. Again,
this does not happen since ~ is a Borel equivalence relation.











Let A C 2" be finite with A ={ao, ...,an-1} ordered lexicographically. Let
D(A) ={d(ai_1, ai) : 1 of A. Further let I C {0,..., n 1} and J C {1,..., n 1}. Then

A: I = {ai:i E I}

is the I-subset of A and

D(A) : J = {d(ajl, a) : j E J}

is the J-subset of D(A).

Theorem 4.2.2. Let -< be a total order on {0,..., n 1}, p E 2n-1 and T be a Cmin
tree. Then for any Borel equivalence relation ~ on [T],,, there is an I C {0,..., n-1}
and J C {1,..., n-1} and a Cmin P C T such that for any x, y E [P]",,, the following
are equivalent:

(1) x y.
(2) x: I = y: I and D(x) : J = D(y) : J.

Proof: For n = 1, this is just the lemma.
For the induction step, use the special constructions from [7]. Let T E [T] p
with T = {ao, ..., -1}<,,- Define a mapping f : {0,..., n 1} -- {0,..., n- 1}
where f(i) is the unique i+j with j E {-1, 1} such that ainai-j is an initial segment
of ai n ai+j.
Definition of T + i.
For i < n let T + i E [7T]"1 result from T by adding a new element at E T
to T such that the following is valid:

(i) at
  • (ii) aci n at is an initial segment of ai n af(i).









    (iii) for j = 0, 1 it holds that if (ai n af(i)) j is an initial segment of at; thus
    a+ leaves af(i) in the same direction as a, does.
    (iv) T and (T \ {ai}) U {a+} have the same shape and parity pattern.
    Definition of T e i.
    For i < n let T ei E [T]"n+ result from T by adding a new element a, E T
    in the following way:
    (i) a? (ii) ai n af(i) is a proper initial segment of ai n cl
    (iii) T and (T \ {aI}) U {af} have the same shape and parity pattern.
    Let the resulting sets T + i resp. T e i have shape - resp. -f and pf. T + i or T E i need not exist for every T [T],p, but certainly
    there exist subsets T E [T]n, such that T + i and T e i exists. If they exists, they
    are in general not unique, while their order-types, and only these will be of interest
    in the following, are independent of the special choice of the added elements.
    Now we start with the induction step. Let n > 2 be given and assume that the
    theorem is valid for all k < n. Let ~ be a Borel equivalence relation on [T],p. For
    every i < n the equivalence relation ~ induces a mapping At : [T]+ {0, 1}
    by
    A (T) = 0 iff T : ({0,..., n} \ {i}) ~ T : ({0,..., n} \ {i + 1}).

    Since the functions At are Borel mappings, by Theorem 3.2.2., there exists a
    Cmin subset To C T and for every i < n there exist a number c E {0, 1} such that
    A([ ToJ]1,+) = c
    Case 1: If this To is such that At([Toj]+ +) = 0 for some i.
    For T e [To]" let the resulting set T : ({0, ..., n 1} \ {i}) have shape -<* and
    parity pattern p*. Then ~ induces a equivalence relation ~* on [T.n- by
    parity pattern p*. Then -, induces a equivalence relation -,* on [*TO-]-.,p by


    (To: ({0,...,n 1} \ {i})) ~* (TI: ({0,..., n 1} \ {i})) iffTo ~ Ti.









    Restricting to a Cin subtree of To if necessary, we may assume that ~* is Borel and
    well-defined.
    By induction hypothesis there exists a Cmin subset T1 c To such that there is
    some I* C {0, ...,n 2} and J* C {1,..., n 2} such that
    n--1
    A ~* B iff A : I* = B : I* and D(A) : J* = D(B) : J* for any A, B E [T1]" <..

    Now for any To, T, E [7],,, let I C {0,..., n 1} and J C {1,...,n- 1} be such that
    for T E [7]",,,, we have that

    (T: ({0,...,n-1} \{i})): I*= T:I

    and
    D(T: ({0,...,n- 1}\{i})): J*=D(T): J.

    It follows for To, T1 e [T1]", that

    To ~ T1 iff To : I = Ti : I and D(To) : J = D(T1) : J.

    Case 2: If To is such that c = 1 for all i:
    For i < n the equivalence relation ~ induces a mapping A : [IT]nl, -

    {0,1} by

    A?(T) = 0 iff (T : ({0, ..., n}) \ {i})) ~ (T : ({0,..., n} \ {i + 1})).

    We may assume that each A is continuous. By Theorem 3.2.2., there exists

    a C,in subtree T1 C To and for each i < n and di E {0, 1} such that

    A? ([T7],) = di.

    Consider for every two-element set {To, T1} [[7] <,]2 the union To U T, say To U
    Ti = p, of shape -<* and parity pattern p*, let Ko, K1 C {0, ..., p 1} be subsets such
    that


    (To U T) : Ko =To and (To U T) : K = T.









    Let (({1, ...,pi}, -<*, p*), K9, Ki')i ring triples. For every i < q let Ai : [Ti]p.,. 2 be a mapping defined by

    A,(T) = 0 iff (T: Kg) ~ (T: Kg).

    Since the functions A, are continuous, by Theorem 3.2.2., we have a ci,
    subtree T2 C T7 such that

    A r[ [T2]',p = c,where ci E {0, 1} for every i.

    For any T E [T2]_,,, let I = {i < n : di = 1}. We claim that for all To, T E
    [T2],,, we have that

    To ~ Ti iff To : I = T : I and D(To) = D(TI).

    For i= 0,1, let Ti = {ai, ..., a~} be given. At first we show the implication
    from right to left. So let To : I = T : I and D(To) = D(Ti). If I = {0,.., n 1},
    there is nothing to prove. So let I {(0,...,n 1} and take I E {0,...,n 1} \I.
    Since dt = 0, we conclude that

    {a 0,..., a~ <, ~ }...~ \ {a} U {a}.

    Iterating this we get

    {ao,...,a o_}<,e, {a : ie o}U {a: j {0,..., n-}\I}

    and thus To ~ T1.
    Now we prove the implication from left to right. Let To ~ T1 and assume
    To : I T : I or D(To) : D(T1). Suppose D(To) $ D(Ti). Let

    d = min((D(To) \ D(Ti)) U (D(T1) \ D(To))),

    where without loss, d = d(a_, a). Since d is minimal, there is j < 2 such that


    a (a9_1 n a9) j is not an initial segment for any / E Ti.









    Let a be an initial segment of, say, ao. Choose &i E T2 \ (To U T1) such that

    (i) (a n &i) -' j is an initial segment of &,.
    (ii) a is not an initial segment of &i.
    (iii) To U Ti and ((To U TI) \ {ao}) U {&i} have the same shape and parity
    pattern.
    Choosing other sets To, T1 if necessary, this is possible. But then we have

    To T1 ~ (To \ {ao}) U {da},

    contradicting the fact that At([To] +) = 1 for every i.
    S(o- Suppose now D(To) = D(T1) but To : I Ti : I. Then there exists i E I with
    a9 $ ail, say af To U T1 and ((To U TI) \ {al}) U {a} have the same shape and parity pattern. As
    before, we get
    To ~ T ~ (T7i \ I{as ) U {a},

    which contradicts the fact that di = 1. This completes the proof of the theorem.














    CHAPTER 5
    HALPERN-LAUCHLI THEOREM
    5.1 Introduction

    In the following we will describe some of the the known facts about the
    Halpern-Liuchli Theorem. We will prove a Cmin version of the Halpern-Liiuchli

    Theorem and an infinite version of the C,in version of the theorem. Please refer to
    Chapter 2 for basic definitions.

    Theorem 5.1.1. ( Halpern-Lauchli) Let d be a natural number, let Ti be a perfect
    tree of height w for every i < d, T= (Ti : i < d), and let k be a natural number. For
    every function f : ( T-+ k there is an A E [w]W and a perfect subtree Ui C Ti for
    every i < d such that the function f [ (A U is constant, or in other words, A U
    is homogeneous for f.

    Let us call the above theorem HLd. In [8], the infinite version of the Halpern-
    Lauchli theorem HL, was proved. It has two other equivalent restatements.

    Theorem 5.1.2. The following are equivalent:
    (i) HL,
    (ii) If P, is the partial ordering for adding r side-by-side Sacks reals with

    countable supports, then every X C w in V1 contains, or is disjoint from, some
    Y E [w]W with Y E V.
    (iii) If fi : i < w are continuous functions from the Hilbert cube [0, 1]" into

    [0, 1], then there exists non-empty perfect sets Pi C [0, 1](i < w) and A E [w]W such
    that, on (i<,, Pi, (fi : i E A) is monotonic.









    5.2 C,,i Tree Case

    Theorem 5.2.1. ( The Ci, version of the Halpern-Liuchli theorem) Let d, k be
    natural numbers, {Ti : i < d} be Cmin trees. Then for any function f : ( Ti --+ k
    there is an A E [w]W and Cmin subtrees Ui C Ti such that f [ (A U= c for some
    c< k.

    The theorem can be easily proved by induction using the following [13]:

    Definition 5.2.2. Let T be a tree. A set A C T is a n-dense set if and only if there
    is an m > n such that A C T(m), the m-th level ofT, and for every s E T(n) there
    exists a t E A such that t D s. Similarly, if To,..., Td- are trees and Ai C Ti(m) for
    all i = 0,..., d 1, then the sequence (Ao, ..., Adi) is n-dense set in (To, ...,Td-1) if
    and only if for every i < d the set Ai is n-dense in Ti.

    Theorem 5.2.3. (Dense set version of Halpern-Liuchli Theorem)
    Let d, k be natural numbers, {T, : i < d} be a set of perfect trees. Then for any

    function f : Ti -+ k, there is an A E [w]W, such that either one of the following
    is true:
    (i) Vn E w, there is an n-dense set (Ao,..., Ad-) in (To, ..., Td-) such that

    f"Ao x ... x Ad-1 = {0}, or
    (ii) There is atE To x ... x Td-1 such that for all n E w there is an n-dense
    set (Bo,...,Bd-1) in (To[to], ...Td-[t-1]) such that f"Bo x ... x Bd-I = {1}.

    Proof of Cmin-Halpern-Liuchli Theorem: We may work on one dimension
    only, because for finite dimension products, the proof is similar. Given any Cmin tree
    T, without loss of generality, let us assume that the first case of the dense set version
    of the Halpern-Liuchli Theorem occurs. First let n = 0, and find no and Ao that
    satisfies the dense set theorem, trim all the nodes that are not comparable with any
    node in Ao, i.e cut all the nodes s with no r E Ao such that r C s or r D s. Now find
    an m so large that every node t in Ao there are at least two branching nodes tI, t2









    above t such that to is even and Iti is odd. Now find nl > m and A1 using Theorem
    5.2.3.. Repeat the process infinitely many times, we easily get a tree T' that contains
    a Cmin-tree. Eventually, the set {ni : i < w} is the set A required in the Cin version
    of the Halpern-Lauchli Theorem.



    Theorem 5.2.4. HL, is true for the Cmin trees.

    Proof: If d < w and m < k < w, X = (Xi : i < d), X C l some trees Ti, let Am= {(Bi : i < d) : for each i, Bi C Xi, CardBi = 4 and each

    Bi = {t^tos, ttos2,trt^sa,^ t^t 4} for some t E Ti(m) and Itol is odd and It|l is
    even }.
    Call S a 4-branch subtree of the Cmi tree T if S is the union of four distinct
    maximal branches in T and such that t^to and t^tl, where tol is even and Itll is odd,
    are the branching nodes above the stem (of the four branches) t.

    Lemma 5.2.5. Let d < w and Ti(i < d) be Cmin trees. Let m < w. Then there
    is p(m,T= (Ti : i < d)) < w, large enough so that for any G C )i G n @B / 0 for all BE A,((Ti(p) : i < d)), then there is an m-dense (Yi : i < d),

    Y C Ti(p), with @YC G.

    Proof: This is a consequence of the dense set version of the Halpern-Laiuchli
    Theorem. If for some m and all p a counterexample Gp,m existed, then UpGp,m would
    be a subset of (T"(Ti : i < d) not containing the product of an m-dense sequence; by
    the dense set version of the Halpern-Liuchli Theorem, there exists t E ) T and for
    infinitely many p, a t-p-dense sequence Xp with & Xp nGp,m = 0. Choose p such
    that each ti E Ti has two splitting nodes t^to and t"tl above it such that Ito0 is even
    and |tll is odd and the splitting nodes are below p; this is a contradiction.









    It suffices for the general case to prove the Cmin version of HL, for those
    sequences (Ti : i < w) of Cmin trees such that

    limit (least level mi where Ti splits) = w.

    Let T7 be the collection of such sequences. For T E T,, an n-dense sequence
    in T is an (Xi : i < w) such that for some m, each Xi C T (m), (Xo,...,Xn-1) is
    n-dense in (To, ..., Tn-1), and Xi 0 0 for i > n. If x E ( T, an x-n-dense sequence
    is an n-dense sequence in (Ti [ i : i < w).

    Lemma 5.2.6. If T E T, A E [w]O, G C (A T and for each n < w there is an
    n-dense X with ( X CG, then there exists Cmin Ti C Ti(i < w) and A' E [A]NO with
    (A' T' CG.

    Proof: This is clear.



    Lemma 5.2.7. If T E T,, A E [w]"O and (A T= Go U G1 then either
    (a) for all m there is an m-dense X with & X C Go, or
    (b) for some non-zero d < w there exists 4-branch subtrees Si of Ti(i < d),
    Cmin subtrees Ti ofTi(i > d), A' E [A]~o, such that
    (*) for any infinite set A" E [A']'o and any sequence (Ti" : i > d), with T" a
    Cmin subtree of T7, there exists t E 9Al" T", say height t= q, such that for any u

    E @i
    Proof: Suppose that (a) fails for m, then m will be the d of (b). Let p be
    the p(d(Ti : i < d)) of lemma 5.2.5.. Thin down each Ti to T(i < d), such that Ti is
    the same as Ti at levels < p, and above level p consists of, for each node t E Ti(p),
    a single maximal branch of Ti above t. Let ao,..., a, enumerate Amn((Ti(p) : i < d)).
    For each p' > p this induces an enumeration ao(p'),...,al(p') of Am((Ti(p') : i < d)).









    Now construct a sequence by induction on 1' < I as far as the construction
    can be continued,
    Vi. y D ...i D..,

    of downwards closed C,mi subtrees of Ti, for each i > d, and a sequence

    BO B D ... D B' D ...,

    with each Bj an infinite subset of A, so that for each 1' < 1, for any tE B (Vl/ :
    i > d) there exists u E (ai, (height t) with u"t E Go.
    This thinning process cannot be continued through all +1 stages. Otherwise
    there would be tE (A Ti : i < d), level t > p such that for all 1' < 1 there exists
    uE (ayl ( height t) with u^t E Go. For u E 1i if u^t E Go. By definition of p, Go would contain an m-dense sequence; appending
    t to it would give an m-dense X in (Ti : i < w) satisfying (a).
    Thus we shall get to a 1' < I where Vi', Bil can be chosen but not Vi'+1, B1'+.
    Let A' = B1', Ti = Vt'(i > d) and let Si(i < d) be the 4-branch subtrees of Ti(i < d)
    determined by ar,+1. These satisfy the property (*).



    Lemma 5.2.8. If T E T, and 0" T= Go U G1, then either
    (a) for all m there is an m-dense X in T with ( X C Go, or
    (b) there are 4-branch subtrees Si of Ti (i < w) and A E [w]'O with A S C G1.

    Proof: By Lemma 5.2.7., pick a non-zero do < w, Ao E [w]N, 4-branch
    Si C Ti (i < do) and Cmin T C0 Ti(i > do) such that for any B E [A]'o and Cin
    Ui C Ti(i > do), (B(Ui : i > do) n Go 0, where we define, for t E Ao(Ui: i > do),
    t E Go if and only if for every u E (&' Also, arrange that there exists te W(To : i > do), tE Go, such that no
    T splits before height (t) by thinning down the TO if necessary; let height (t)= ao,









    the first member of A. By the assumption that (a) fails, there is m such that no
    m-dense sequence X in (To : i > do), with height X E Ao, satisfies ()XnG G 0.
    Thus we may carry out the construction again with respect to (TO : i > do), Ao
    and G; that is, using lemma 5.2.7, find 4-branch Si C To(do < i < di), Cmmn tree
    Ti C To(i > di), A' E [Ao0]N, and GI(te G1 if for every uE <9i level as t, u^tE GO), with )B(Ui : i > di) n G\ : 0 whenever each Ui is Cmi in Tl
    and B E [A']~o, and thin out to pick al > ao as before. Repeat this construction for
    infinitely many times, getting (Si : i < w) and A = {an : n < w}; these satisfy the
    lemma.


    To prove the theorem, we are given "& T= Go U G1, and will construct Cmin
    trees Ui(i < w) and a map from (" U into (" T, and show that a homogeneous

    (A S (Si a 4-branch subtree of Ui) induces a homogeneous (B T' (T' a Cmin subtree
    of T).
    We may assume that for each i,

    Ti = {s E 2<' : s(j) = 0 Vj < i} = Ki,

    for (Ki : i < w) is isomorphic to a sequence of subtrees of (Ti : i < w) restricted to
    a common infinite set of levels. Let (B, : n E w) be a disjoint partition of w, with
    each Bn infinite. We may assume that each B, has infinitely many odd numbers and
    even numbers in it. Now, via some bijection between w and w x w, {Uj : j < w} will
    be {Ui,n : i, n < w}, where

    Ui,n := {s E 2< : s(j) = 0 whenever j V Bn or j < i}.

    Then, if u E0W U, height u =m, define H(u) =t EQ" T, height t =m by
    the rule:

    S ui,, if i j < m, j E Bn,
    t0j) o if j < i.









    Then H is a level preserving bijection between (& U and & T. We have now
    0W U= Go U G1, where Gj := H-'"Gj. By the last lemma, there exist an infinite
    set A C w and 4-branch subtrees Si,n of Ui,(i, n < w) and j < 2 with (&A SC Gj.
    Then there are downwards closed subtrees T' of T, with H" &" T', and we have
    (A T'C Gj. It is clear that each Ti is Ci. This completes the proof. O

    Definition 5.2.9. A nonprincipal ultrafilter U on w is selective if for every sequence
    (An : n < w) of elements of U there is A E U such that A \ n c An for every n E A.
    Such A is said to be the diagonalization of the sequence (An : n < w).

    Lemma 5.2.10. A non principal ultrafilter U on w is selective if for every partition
    p : [w]2 -+ 2 there exists i < 2 and A E U such that p"[A]2 = {i}.

    Lemma 5.2.11. If U is a selective ultrafilter on w and if A is an analytic subset of

    [w]W, then there is an A E U such that [A]W C A or [A]" n A = 0.

    Definition 5.2.12. IfU is an ultrafilter on w, let HLd(U) be HLd with the require-
    ment that the set A be a member of U.

    Theorem 5.2.13. For every d < w and every selective ultrafilter U on w, the Cin
    version of HLd(U) is true.

    Proof: Let Ti(i < d) be a given sequence of Cmin-trees and let p : 0i
    {0, 1} be a given coloring. If d = w, we further assume, without loss of generality,
    that if si is the minimal splitting node of Ti for i < w, then the length of s, increases
    with i. Let

    A = {A E [w] : (Vi < d)(3 Cmin Ui C Ti)(3e < 2)p" (& Ui A = {}}.
    i Then A is clearly an analytic subset of [w]" so by fact 2, there exists A E U
    such that either [A]W C A or [A]W n A = 0. Note that by the Cmin version of
    Halpern-Lluchli theorem, the second alternative does not happen, so we are done.









    Theorem 5.2.14. Any finite product of Cmin forcing preserves the selective ultrafil-
    ter, hence does not add any independent sets.

    This follows from the next theorem.

    Theorem 5.2.15. Let P be the Cmin forcing. Every selective ultrafilter U generates

    a selective ultrafilter U* in the forcing extension VPW.

    Proof: The only thing left to prove is that

    U* = {A C w : 3B E U such that B c A} is selective;

    that is, every sequence (An : n < w) of elements of U* has a diagonalization in U*.
    But this is easy to show. Consider a fusion sequence P, where s E 2<, i < w, such

    that for every n < w and s= (si : i < w) in (2n)W the condition (P, : i < w) forces
    that An is equal to some element Bs of U, we may do so since we can without loss

    of generality assume that each An is taken from U. Let (Pi : i < w) be the fusion of

    this sequence and let B E U be such that B \ n c B, for every n E B and sE (2n)".

    This finishes the proof.


    We also have the Cmin analogue of the perfect set result on Hilbert cube:

    Theorem 5.2.16. For every sequence {fn : n < w} of continuous functions from the

    Hilbert cube [0,1] into the interval [0,1], there is a subsequence {f'} and a sequence

    {P,} of Cmin subsets of [0, 1] such that {f/} monotonically converges to a continuous
    function on the product Po x P1 x P2....

    Proof: Let P be the Cin forcing. In the forcing extension of P" look at the
    sequence of reals f,((si : i < w)), where (si) is the canonical name for the generic
    sequence of Cmin reals. Under the assumption that there is a selective ultrafilter U,
    U* is a selective ultrafilter, apply the above fact to the partition, p : [w]2 2 defined









    by

    p({m, n}) = 0 iff f,((si : i < w)) < fm((s : i < w)),

    and get a member A of U* such that fn((si : i < w)) (n E A) is monotonic. Fix
    (Pi : i < w) E PW deciding the set A and the fact that the sequence is, say, increasing.
    Let

    F = {(x : i < w) E fPi : (Vm < n E A)fm((i : i < w)) < fn((x : i < w))}.
    i Notice that F is a closed subset of 1fi<, Pi. Since every nonempty open subset
    of this product contains a product of a sequence (Pf : i < w) of Cin subsets which
    must also force that the sequence fn((si : i < w)) is increasing, the set F must be
    equal to the product 1i<, Pi. Let g : i<, Pi -+ [0,1] be the limit of fn(n E A).

    Since g is Baire, by further restricting to the subtrees if necessary, we may assume it
    is continuous. So we are done.



    Theorem 5.2.17. For every finite d and a sequence {fn} of continuous functions
    from the cube [0, 1]d into [0, 1], there is a single Cmin set P C [0, 1] and a subsequence

    {fn } which is monotonically convergent on pd.

    Proof: The set P will be the result of a fusion sequence P,(s E 2 structed together with a decreasing sequence of set An(n < w) such that for every
    n < w and every sequence si of distinct elements of 2" the subsequence {fk :k E An}
    when restricted to the product I-i function. Clearly, there is no problem in constructing this fusion using the previous
    result. It should be pointed out that if {kn}n<, is a strictly increasing sequence such
    that kn E An for every n < w then {Fk }n<, and the set P satisfies the conclusion of
    the theorem.















    CHAPTER 6
    OTHER FORCING NOTIONS

    6.1 The En Trees

    Let Eo be the relation on R, such that


    for any x, y E R, x ~ y iff x and y differ by a rational.

    It is easy to see that E0 is indeed an equivalence relation. When working with

    the space 2W, let x ~ y mean that x(n) = y(n) for all but finitely many n.

    Let I be the ideal a-generated by the partial Borel Eo-selectors, where a partial

    selector is a subset of R which meets each equivalence class in exactly one point. It

    is well-known that Borel Eo selectors are meager and it is easy to show that any

    member in the E0 ideal I has Lebesgue measure 0, hence Eo is nontrivial. For more

    information about the E0 ideal, see [14].

    Definition 6.1.1. An Eo-tree is a non empty tree T C 2W which is perfect, that is,

    every node in T can be extended into a split node, and homogeneous, i.e., for every

    split node t E T there are two longer split nodes so, sl of equal length such that every

    node compatible with t is compatible with so or si and {u : s'o E T} = {u : su E T}

    Theorem 6.1.2. [14] Every analytic subset of 2W is either in the ideal I or it contains

    all branches of some Eo-tree.

    The following result is due to Blass in a private conversation:

    Theorem 6.1.3. There is a Borel function f : [2"]3 2W such that for any Eo tree

    S C 2", we have that f"[S]3 = 2".









    Proof: Define a Borel function f from [T]3 to 2', where T is any Eo-tree,
    such that for any E0 subtree S C T we have that f([S]3) = 2". Given any set

    {x1,x2, x3} ordered lexicographically, we may assume that all of the pairs (xi, j)
    differ at infinitely many places for all 1 < i < j < 3. Let

    m(xi, 2) = 3, m(x2,x3) = 1, m(x1, 3) = 2.

    Since we are working on 2", there must be a least natural number ko and a
    pair (x, x) selected from the set {xIx2, x3} such that x (ko) = x (ko); for n > 1 let
    the natural number kn be the n-th smallest number such that for some pair (x!, xz),
    we have that x'(kn) = x{(kn) and such that the pair (x' x) (x_-1 ,-1). Define
    f(,(xi,, 23)(n) to be Im(xz, xZ) m(xZ, xn)I mod 2, where (xy, xn), (xn,xn) are such
    that x](kn) = xz(kn) and x'(kn+1) = xn(kn+l)
    Observe that if we divide 2W into m disjoint open balls, we get a function that
    will hit every partition class for any E0 tree S C 2. Hence no partition theorem is
    possible.

    6.2 The Silver Forcing

    In the following we study the Silver forcing (See [14] for more detail). The
    Silver ideal is a variation of the Eo-ideal. Let G be the graph on 2W connecting two
    binary sequences if and only if they differ in exactly one place. Let I be the ideal
    o-generated by Borel G-independent sets.

    Definition 6.2.1. A Silver tree is a nonempty tree T C 2W which is perfect, and
    homogeneous in the sense that for every split node t E T every node compatible with
    t is compatible with t^O or t^l, and {u : t^0u E T} = {u : tl"1u E T}.

    Theorem 6.2.2. [14] Every analytic subset of 2W is either in the ideal I or it contains
    all branches of some Silver tree.









    The following results were generalized by Blass in a private conversation after
    I told him how to construct a continuous function f : [2W]2 -+ 2 such that for any
    Silver tree S we have that f"[S]2 = 2.

    Theorem 6.2.3. There is a continuous function f : [T]n -+ m, where T is any Silver
    tree, such that for any Silver tree S C T we have that f" [S] = m.

    Proof: Define a continuous function f from [T]" to n, where T is a Silver
    tree, the following way:
    Given a Silver tree T and {xi : i < n} set of elements of T, let s be the lowest
    branching point for the set {xi : i < n}, and let k be the number of l's in s modulo
    m. Define f({(x : i < n}) = k.
    It is easy to check that no homogeneous Silver subtree S C T for f is possible.



    Theorem 6.2.4. There is a continuous function f : [T]" -+ w, where T is any Silver
    tree, such that for any Silver tree S C T we have that f" [S]'" = w.

    Proof: Given any Silver tree T and n-set {zi : i < n}, let s be the lowest
    branching point of the n-set as before, now let k be the largest number m such that
    2" divides the number of l's in s.
    It is easy to check that this definition works.



    Theorem 6.2.5. There is a Borel function f : [T]2 2", where T is any Silver tree,
    such that for any Silver subtree S C T we have that f"[S]2 = 2".

    Proof: Given any xZ, x2 E T, without loss we may assume that x1, x2 differ
    at infinitely many places. Define the function f by induction: At the n-th stage look
    at the n-th point that x1, x2 differ, if it is the case that xl(n) = 0, x2(n) = 1 then let

    f(x1, z2)(n) = 0, otherwise, f(x1,x2)(n) = 1. If x1,x2 differ by only finitely many,
    say, n places, let f(xl, x2)(m) = 0, whenever m > n.











    6.3 The Packing Measures

    In the following we define the packing measure on the space 2" with the metric

    d(x, y) = 1r, where n is the least number such that x(n) 5 y(n). For any positive
    real number h < 1, any set A C 2W, we say that {(xi, ri) : xi E A} is a 6-packing on

    A if the open balls containing xi with radius ri are disjoint and 2ri < 6 for any i. Let


    Po,(A) =sup{() rh : {(xa, r,) : i < w} forms a 6-packing }.

    let

    Po^(A) = limasoPOh6(A),

    and define the packing h-measure for A C 2W by
    00
    ph(A) = inf{- Poh(Ai) : A C UiAi}.
    i=1
    Let Ih be the a -ideal generated by the subsets of 2" with finite packing h-

    measure.

    Theorem 6.3.1. For any perfect tree T, there is a continuous function f : [T]3 -+ 2,

    such that for any Ih-positive S C T, we have that f" [S]3 = 2.

    Proof: Fix a sequence of partitions (fn : n < w) such that each fn : [n]2 -+ 2

    and each f, witness the smallest possible size of homogeneous set. By a combinatorial

    theorem [2], which stated that

    -k 2k 2\ k'


    where nk is the least natural number such that for any partition f : [nk]2 -+ 2

    there exists a homogeneous subset of size k, let dn be the smallest number such that

    v/d2" > n. Given any {zo, x2} E [T]3, we will get exactly 2 forks, where a fork is an

    s E 2<' such that both s'0 and s^l are initial segments of some xi, where i = 0, 1, 2.









    Go to the level of the tree where the highest of the 2 forks are located, say, level n.
    We have exactly 2 nodes, say s, t E 2n that are initial segments of xo, x1 or x2. Now
    look at all the nodes at level n and let us say that we have totally m elements on
    the n-th level of the tree ordered lexicographically and say that s is the i-th element
    and t is the j-th element of the n-th level of the tree. Let f(xo, X1, x2) = fm(i, j).
    Now it is easy to see that under this definition f is really continuous. We can easily
    compute the weight at the n-th level, which is

    S(( )n+1)h =m (( 1)n+l)h = a

    The biggest size of homogeneous set that we are able to obtain is less than dn. Now
    the size of the homogeneous set in the next level is at most d +2 by the last theorem.
    So the weight on the next level is < (dn + 2) a (I)h. So the sum of all the weights
    at all the levels is easily seen to converge to a finite value using the ratio test. Hence
    it is impossible to have a Ih-positive subset that is homogeneous.















    REFERENCES


    [1] A. BLASS, A partition theorem for perfect sets, Proc. Amer. Soc., 82 (1981),
    pp. 271-277.
    [2] M. BONA, A walk through combinatorics, World Scientific, Hackensack, 2002.

    [3] P. ERD6S AND R. RADO, A combinatorial theorem, J. London. Math. Soc.,
    25 (1950), pp. 249-255.
    [4] S. GESCHKE, M. KOJMAN, W. KUBIS, AND R. SCHIPPERUS, Convex decom-
    positions in the plane and continuous pair coloring of the irrationals, Israel J.
    Math., 131 (2002), pp. 285-317.
    [5] F. GALVIN AND S. SHELAH, Some counterexamples in the partition calculus,
    J. Combinatorial Theory Ser., A15 (1973), pp. 167-174.
    [6] J. D. HALPERN AND H. LAUCHLI, A partition Theorem, Trans. Amer. Math.
    Soc., 124 (1966), pp. 360-367.
    [7] H. LEFMANN, Canonical partition behaviour of cantor spaces, in Irregularities
    of Partitions, edited by G. Halisz and V.T. S6s, Springer-Verlag, New York,
    1989, 93-105.
    [8] R. LAVER, Products of infinitely many perfect trees, J. London Math. Soc., 29
    (1984), pp. 385-396.
    [9] J. MYCIELSKI, Independent sets in topological algebras, Fund. Math., 55 (1989),
    pp. 139-147.
    [10] J. MYCIELSKI, Algebraic independence and measure, Fund. Math., 61 (1967),
    pp. 165-169.
    [11] F. P. RAMSEY, On a problem of formal logic, Proc. London Math. Soc., 30
    (1930), pp. 264-286.
    [12] S. SHELAH, AND J. ZAPLETAL, Duality and the pcf theory, Mathematical
    Research Letters, 9 (2002), pp. 585-595.
    [13] S. TODORCHEVICH AND I. FARAH, Some application of the method of forcing,
    Yenisei, Moscow, 1995.
    [14] J. ZAPLETAL Descriptive set theory and definable forcing, AMS Memoirs, Prov-
    idence, 2004.















    BIOGRAPHICAL SKETCH


    Yuan-Chyuan was born in Taiwan in 1973. He spent the first 25 years of his

    life on that island. He came to Gainesville to study mathematics in 1998. In 2001

    he began studying under Professor Jindrich Zapletal and working toward a doctoral

    degree.









    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of actor of Philophy.


    Jn rich Zapletal Phairman
    instant Professor of Mathematics

    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of Philosophy.


    Jean A. Larson
    Professor of Mathematics

    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of ilosop


    WifAm Mitc nlI
    Professor of Mathematics

    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of Philosophy.


    Rick L. Smith
    Associate Professor of Mathematics

    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of Philosop.


    Greg Ray
    Associate Professor of Philosophy

    This dissertation was submitted to the Graduate Faculty of the Department of
    Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
    and was accepted as partial fulfillment of the requirements for the degree of Doctor
    of Philosophy.

    May 2005
    Dean, Graduate School







































    UNIVERSITY OF FLORIDA
    1lU1111 III l IBII III I III II 111112 I
    3 1262 08554 2412


    .S 5se




  • Full Text
    PARTITION PROPERTIES AND HALPERN-LAUCHLI THEOREM ON THE
    Cmin FORCING
    By
    YUAN-CHYUAN SHEÜ
    A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
    OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
    OF THE REQUIREMENTS FOR THE DEGREE OF
    DOCTOR OF PHILOSOPHY
    UNIVERSITY OF FLORIDA
    2005

    ACKNOWLEDGEMENTS
    I would like to thank my advisor, Jindrich Zapletal, for his patience and almost
    infinite discussion with me on the problems I was trying to solve. Without him, I
    simply could not have got this far.
    I would also like to thank the other members of my supervisory committee:
    Dr. Jean A. Larson, Dr. William Mitchell, Dr. Rick L. Smith, and Dr. Greg Ray. I
    have learned a lot from them either through taking classes taught by them, personal
    conversations, or hearing them speak in seminars and conferences.
    I am very grateful to the Department of Mathematics for supporting me with
    a teaching assistantship during my years as a graduate student. I also would like
    to thank the College of Liberal Arts and Science for supporting me with a Keene
    Dissertation Fellowship in the spring 2005 term.
    Finally, I must acknowledge the contribution of my family and all my friends
    in the Department of Mathematics for their caring and support.
    n

    TABLE OF CONTENTS
    ACKNOWLEDGEMENTS ii
    ABSTRACT iv
    CHAPTERS
    1 INTRODUCTION 1
    2 PRELIMINARIES 3
    2.1 Basic Definitions 3
    2.2 The Property of Baire 5
    2.3 The space 2U 6
    2.4 The Cjnin Trees 6
    2.5 Determinacy 7
    3 BLASS PARTITION THEOREM 8
    3.1 Blass Theorem on the Perfect Sets 8
    3.2 Partition Theorem on the Cmin Tress 10
    4 CANONICAL PARTITION THEOREM 17
    4.1 Perfect Tree Case 17
    4.2 Cmin Tree Case 18
    5 HALPERN-LÁUCHLI THEOREM 24
    5.1 Introduction 24
    5-2 Cjnin Tree Case 25
    6 OTHER FORCING NOTIONS 33
    6.1 The E0 Trees 33
    6.2 The Silver Forcing 34
    6.3 The Packing Measures 36
    REFERENCES 38
    BIOGRAPHICAL SKETCH 39
    iii

    Abstract of Dissertation Presented to the Graduate School
    of the University of Florida in Partial Fulfillment of the
    Requirements for the Degree of Doctor of Philosophy
    PARTITION PROPERTIES AND HALPERN-LÁUCHLI THEOREM ON THE
    Cmin FORCING
    By
    Yuan-Chyuan Sheu
    May 2005
    Chairman: Jindrich Zapletal
    Major Department: Mathematics
    In this dissertation, we focus on a special breed of forcing notion called Cmin
    forcing, which is forcing equivalent to the form R(R) \ I for some Cmi„-ideal I.
    We define a special perfect subsets of the space 2W, called Cmin trees. We study
    the partition properties and Halpern-Láuchli theorems on these trees. We obtained
    three principal theorems which will be our dicussion Chapters 3,4 and 5 respectively.
    We also show that the some other forcing notions do not have similar partition
    properties. This is in Chapter 6.
    IV

    CHAPTER 1
    INTRODUCTION
    This dissertation is based upon the deep combinatorial theorem of Halpern-
    Láuchli [6] on the perfect trees which Halpern and Láuchli developed in the 1960’s.
    The Halpern-Láuchli Theorem is obtained as a byproduct of the proof of Con(ZF +
    ->AC + BP), where BP is the statement that there exists a prime ideal in every
    Boolean algebra, a consequence of the Axiom of Choice equivalent to Compactness
    Theorem for the First-Order Predicate Calculus. It has been noticed since then that
    the lemma of the proof might be of independent interest.
    One of the main applications of the Halpern-Láuchli Theorem is the proof of
    the well known Blass Theorem [1] on perfect trees, which states
    T ->â–  [5']"Wi(n_1)!
    Let a Cmin tree [14] be a nonempty binary tree T C 2 t G T can be extended both into a splitnode of even length and into a splitnode
    of odd length. The study of the Cmin trees was originated from Geschke et al. [4],
    where the authors defined the Cmin function (see Chapter 2 for definition). The Cmin
    cr-ideal / generated by the Cmin homogeneous sets is nontrivial and the subsets of IP
    that contain all the branches of a Crmn tree are the typical /-positive sets. The Cm¿n
    trees as a forcing notion ordered by inclusion is proper.
    We have shown that for the Cmin trees we have a similar but different
    T ->■ [5]"Wi2„-1 ><(„_!)!
    This is contained in Chapter 3.
    1

    2
    In 1930 Ramsey [11] proved the famous pigeon-hole principle for finite sets. A
    new situation arises if partitions into an arbitrary number of classes are considered.
    For this case, Erdos and Rado [3] proved the so-called canonical version of Ramsey
    Theorem.
    Based on the Cmin version of Blass Theorem, we also proved the canonical
    theorem on Borel equivalence relations on Cmin trees. This means that, fixing a
    shape and parity pattern, there exists a finite set G of equivalence relations on [2w]n
    such that for any Borel equivalence relation / on [2w]n there is an equivalence relation
    g G G and a Cm¿„ tree T C 2“ such that / \ [T]n = g \ [T]". This is in Chapter 4.
    We obtained a Cm¿„ tree version of the Halpern-Láuchli Theorem. In a 1984
    paper, Richard Laver [8] generalized the existing Halpern-Láuchli Theorem to the
    infinite product of perfect trees. From there, equivalent results on Hilbert cubes and
    selective ultrafilters can be derived. We have developed the Halpern-Láuchli Theorem
    on the infinite product of Cmin trees and we have the equivalent results on Hilbert
    cubes and selective ultrafilters as well. This is contained in Chapter 5.
    We have also shown that the Cmin trees are special by constructing counterex¬
    amples that demonstrate that no such results are possible for a variety of other kinds
    of forcing notions such as E0 Forcing, Silver Forcing and the forcing notion associated
    with the Packing Measure. This is the main content of Chapter 6.

    CHAPTER 2
    PRELIMINARIES
    2.1 Basic Definitions
    A relation is a set R all of whose elements are ordered pairs. / is a function
    iff / is a relation and
    Vx G dom(/) 3ly G ran(/) ((x,y) G /).
    / : A —> B means / is a function, A = dom(/), and ran(/) C B. If f : A —> B
    and x G SR, the set of all real numbers, /(x) is the unique y such that (x,y) G /;
    if C C A, / \ C = f DC x B is the restriction of / to C, and f"C — ran(/ f C)
    = {f{x) -xeC}.
    A total ordering is a pair (A, R) such that R totally orders A; that is, A is a
    set, R is a relation, R is transitive on A:
    Vx, y,z G A (xRy and yRz —> xRz),
    trichotomy holds:
    Va:, y G A(x = y or xRy or yRx),
    and R is irreflexive:
    Vx G A it is not the case that xRx.
    As usual, we write xRy for (x, y) G R.
    We say that R well-orders A, or (A, R) is a well-ordering iff (A, R) is a total
    ordering and every non-empty subset of A has an R-least element.
    Definition 2.1.1. (a) A partial order is a pair {P, <)such that IP 7^ 0 and < is a
    relation on IP which is transitive and reflexive (Vp G P(p < p)). p < q is read “ p
    extends q Elements of P are called conditions.
    3

    4
    (b) (P, <) is a partial order in the strict sense iff it in addition satisfies
    Vp, q(p < q and q < p —>• p = q). In that case, define p < q iff p < q and p ^ q.
    Definition 2.1.2. An equivalence relation on a set X is a binary relation which
    is reflexive, symmetric, and transitive: For all x,y,z € X, we have that x ~ x,
    x ~ y —> y ~ x, if x ~ y and y ~ z then x ~ z.
    Definition 2.1.3. A tree is a partial order in the strict sense (T,<), such that for
    each x E T, {y eT : y < x} is well-ordered by <.
    The set of all finite sequence of 0’s and l’s, 2 s < t iff s C t. For any T C 2 T. Any T C 2 Definition 2.1.4. A skew tree is a tree so that on each level there is at most one
    branching node.
    For distinct a, e [T], let d(a, ¡3) be the level of the highest common node
    of the paths a and ¡3 through T. Here we assume for any rc-set it is the case that
    the elements are listed in lexicographic order, and we may assume that the trees are
    skew.
    Definition 2.1.5. By the shape of an n-element set {ao,..., on-i} C [T\, we mean
    the linear ordering -< of {1,n — 1} given by i -< j -4=> d(aj_i, ctj) is in a lower level
    than d(aj_i, ctj).
    Definition 2.1.6. By the parity pattern p, of an n-element set {a¡o> Q¡n-i} £ [T]n,
    we mean p e 2n_1 given by p(i) — 0 iff the i-th (counting from 0) lowest splitnode of
    the n-set is on the even level.
    Definition 2.1.7. Let T — (Tt : i < d) be a seqence of trees. Define
    to be the set of all n-tuples X from the product of {T¿ : i < d} such that |AT(i)| =

    5
    \X(j)\ for any i,j < d; for A C u>, let (%)A T be the set of all n-tuples Y from the
    product of {Ti : i < d} such that |y(¿)| = |y(j)| G A for any i,j < d.
    Definition 2.1.8. An algebra of sets is a collection S of subsets of a given set S
    such that
    (i) S (ii) if X G S and Y G S then X U Y G S,
    (in) if X G S then S\X G S.
    A a-algebra is additionally closed under countable unions:
    (iv) If Xn G S for all n, then U^L0X„ G S.
    For any collection X of subsets of S there is a smallest a-algebra S such that
    S D X; namely the intersection of all a-algebras S of subsets of S for which X C S.
    Definition 2.1.9. A set of reals B is Borel if it belongs to the smallest a-algebra B
    of sets of reals that contains all open sets.
    2.2 The Property of Baire
    Let us call a set A C 2“ nowhere dense if the complement of A contains a
    dense open set. Note that A is nowhere dense just in case that for every non empty
    open set G, there is a nonempty open set H C G such that A fl H = 0. A set A is
    nowhere dense if and only if its closure A is nowhere dense.
    A set A C 2W is meager if A is the union of countably many nowhere dense
    sets.
    Definition 2.2.1. A set A has the Baire property if there exists an open set G such
    that AAG = {A \ G) U (G \ A) is meager.
    Clearly, every meager set has the Baire property. Note that if G is open, then
    G\G is nowhere dense. Hence if AAG is meager then (2U \ A) A(2W \ G) = AAG is
    meager, and it follows that the complement of a set with the Baire property also has

    6
    the Baire property. It is also easy to see that the union of countably many sets with
    the Baire property has the Baire property and we have:
    Lemma 2.2.2. The sets having the Baire property form a a-algebra; hence every
    Borel set has the Baire property.
    2.3 The space 2u
    Let u} be the set of all natural numbers. The space 2U is the set of all infinite
    sequences of numbers 0 or 1, (an : n G u>), with the following topology: For every
    finite sequence s = (a* : k < n), let
    O(s) = {/ € 2“ : s C /} = {(c* : k € uf) : (VA: < n)ck - a*}.
    The sets O(s) form a basis for the topology of 2W. Note that each O(s) is also closed.
    The space 2W is separable and is metrizable: consider the metric
    d(/,j)=2¿T.
    where n is the least number such that f(n) ^ g(n). The countable set of all eventually
    constant sequences is dense in 2W. This separable metric space is complete, as every
    Cauchy sequence converges.
    2.4 The Cmin Trees
    Definition 2.4.1. Let Cmin '■ [2W]2 —> 2 be the mapping defined by
    Cmin{x,y) = A(ar,y) mod 2,
    where A(x,y) is the least number n such that x(n) ± y{n), and let I, the Cmin-ideal
    be the o-ideal cr-generated by the Cmin homogeneous sets.
    The above definition can be found in Geschke et al. [4]. It is not difficult to
    verify that Cmm homogeneous sets must be meager. It is so because if A is Cmin-
    homogeneous, say Vx, y £ A Cmin(x, y) — 0, then all the split nodes of A are on even
    levels.

    7
    Now, it is clear that A is meager. And hence the ideal I is non-trivial.
    Definition 2.4.2. A Cmin tree is a nonempty tree T C 2 t G T can be extended both into a splitnode of even length and into a splitnode of odd
    length.
    The following theorem can be found in Shelah and Zapletal [12].
    Theorem 2.4.3. Every analytic subset of 2U is either in the ideal I or it contains
    all branches of some Cm¿n tree. Under AD (see below) this extends to all subsets of
    2W.
    2.5 Determinacv
    With each subset A of 2U we associate the following game Ga, played by two
    players I and II. First I chooses a number a0 G {0,1}, then II chooses a number
    b0 G {0,1}, then I chooses aj, then II chooses 6i and so on. The game ends after
    u steps; if the resulting sequence (a0, bi, oi, bi,...) is in A, then I wins; otherwise II
    wins.
    A strategy (for / or II) is a rule that tells the player what move to make
    depending on the previous moves of both players. A strategy is a winning strategy
    if the player who follows it always wins. The game Ga is determined if one of the
    players has a winning strategy.
    The Axiom of Determinacy (AD) states that for every A C 2U, the game Ga
    is determined.

    CHAPTER 3
    BLASS PARTITION THEOREM
    3.1 Blass Theorem on the Perfect Sets
    Let 2^ be ordered lexicographically. For X C 2W, let [AT]n be the set of n-
    element subsets of X. When we describe a finite subset of 2u by listing its elements,
    we always assume that they are listed in increasing order. Thus [2w]n is identified
    with a subset of the product space (2w)n, from which it inherits its topology. A subset
    A of 2UJ is perfect if it is nonempty and closed and has no isolated points. This is
    equivalent to saying that A = [T], where T is a tree such that for every t G T there
    exist So>si 5 L both in T, that are incomparable, i.e., neither s0 D Si nor si D s0.
    A. Blass [1] proved the following
    Theorem 3.1.1. Let P be a perfect subset of 2w and let [P]n be partitioned into
    a finite number of open pieces. Then there is a perfect set Q C P such that [Q]n
    intersects at most (n — 1)! of the pieces.
    It was pointed out in Blass [1], that the theorem remains true if 2U is replaced
    by the real line R with its usual topology and order. To see this, it suffices to
    observe that every perfect subset of R has a subset homeomorphic to 2W via an order¬
    preserving map and that any one-to-one continuous image in R of a perfect subset
    of 2W is perfect in R. Second, the hypothesis that the pieces of the partition are
    open can be greatly relaxed. Mycielski [9, 10] has shown that any meager set or
    any set of measure zero in [R]n is disjoint from [P]n for some perfect P C R. For
    the meager case, he obtains the same result with R replaced by any complete metric
    space X without isolated points. It follows that, if [R]n is partitioned into finitely
    many pieces that have the Baire property, then their intersections with [P]n are open
    8

    9
    for some perfect P. Similarly, if the pieces are Lebesgue measurable, they become Gg
    sets when restricted to [P']n for suitable perfect tree P'; since Gg sets have the Baire
    property, we can apply the preceding sentence, with P' as X, to get a perfect P C P'
    such that the pieces intersected with [P]71 are open in [P]n. Thus the theorem, as
    extended by the first remark as above, implies the following
    Corollary 3.1.2. If [R]n is partitioned into finitely many pieces that all have the
    Baire property or are all measurable, then there is a perfect set Q C R such that [Q]7*
    meets at most (n — 1)! of the pieces.
    Some hypothesis about the pieces is necessary, however, for Galvin and Shelah
    [5] have shown that there is a partition of into infinitely many pieces such that,
    for any Q C 2W of the cardinality of the continuum, [Q]2 intersects all the pieces.
    As mentioned in Blass [1], we may assume that the all the perfect trees are
    skew (Definition 2.1.4.). Since -< (see Definition 2.1.5.) tells us in what order the
    paths ai split apart as we proceed up the tree and there are (n — 1)! linear orderings
    -< of {1,..., n — 1} and all of them are obviously realized as shapes within any perfect
    subset of [T], we see that the (n — 1)! in the theorem is optimal.
    We shall need a few definitions: Let T= (T0,..., Tr_i) be a skew r-tuple of
    perfect trees, which means not just each tree Tj is skew but also that no two distinct
    Tf s have splitting nodes at the same level. Let n= (no,nr_i) be a r-tuple of
    positive integers with sum n. By an n-set in T we mean an r-tuple cr whose ¿th
    entry Oi is an n*-element subset of (T¿). The shape -< of such an n-set cr is the linear
    ordering of the pairs (i, j), with 0 < i < r and 1 < j < n¿, given by the levels of
    d(aij-i, a¿¿), where a^j is the j-th element of cr¿ in lexicographic order. The above
    theorem is a special case when r — 1 of the more generalized theorem.
    Theorem 3.1.3. (Polarized Theorem) Let r, T, n, n be as above and let -< be any
    shape of n-sets. Let the collection of all n-sets in T be partitioned into finitely many

    10
    open pieces. Then there exist perfect trees T[ C T¡ such that all n-sets in T' with
    shape X lie in the same piece of the partition.
    3.2 Partition Theorem on the Cm.v, Tress
    Lemma 3.2.1. Every Cm¿n-íree has a Cmin skew subtree.
    Proof: Given any Cm¿n-tree T. To obtain a skew subtree, we must make sure
    that at each level, there is at most one branching node. Also, we have to make sure
    that for each node i 6 T, there are branching extensions to,t\ 6 T of t with |t0| even
    and |f11 odd. We will construct the Cm¿n skew subtree V of T by induction on the
    number of branching nodes below the nodes we are currently looking at, each stage
    make sure the above two requirements are satisfied:
    Suppose we have done up to stage n. At stage n +1, look at all the maximal
    (meaning there is no node extends it so far in the construction) nodes constructed so
    far and make sure each of them has two extensions of branching nodes, one of them
    has even length, one of them has odd length, and so that at each level there is at
    most one branching node. Now it is easy to see that V is a Cm¿n subtree of T.
    â–¡
    The parity pattern p for an n-set cr in a sequence of trees T is defined by
    Pj(i) = 0 iff the ¿-th lowest splitnode in cr, is on the even level.
    Theorem 3.2.2. (Polarized Theorem for the Cmin trees) Let T=< T0,Ti, ...,Tr_i >
    be a set of Cmin trees, n=< no,...,nr_i >, and let X be any shape of n-sets, p be
    a parity pattern of n-sets. Let the collection of all n-sets in T be partitioned into
    finitely many open pieces. Then there exist Cmin trees T[ C Tt such that all n-sets in
    T' with shape x and parity pattern p lie in the same piece of the partition.
    It easily follows from the theorem as a special case when r = 1, we also have
    the following corollary.

    11
    Corollary 3.2.3. Let T be a Cmin-tree, and let [T]n be partitioned into a finite number
    of open( in [T]n) pieces. Then there is a Cmin-tree S C T such that [5]" intersects at
    most 2n_1 x (n — 1)! of the pieces.
    Note that the 2n_1 x (n — 1)! in the conclusion of the corollary is optimal since
    there are (n — 1)! many shapes and 2n_1 many parity patterns for the sets B G [T]",
    where T is a Cm¿n tree.
    Before we prove the theorem, let me point out that it is easy to show (using
    a typical fusion argument; see the theorem below) that any meager set in [2w]n is
    disjoint from some [P]n for some Cm¿n-tree P. Hence we can generalize the theorem
    so that the partition is Baire. The proof follows the original proof of Blass on perfect
    sets.
    Theorem 3.2.4. Given a Cmin-tree T, a natural number m and a meager set A C
    [T]m, there exists a Cmin subtree P of T, such that [P]m fl A = 0.
    Proof: Before we prove the theorem, we need some definitions. The proof
    uses the technique of fusion. Let p be a Cm¿n-tree. A node s G p is a splitting node if
    both s~0 and s~l are both in p; a splitting node s is an n-th splitting node if there
    are exactly n splitting nodes t such that t C s. For each n > 1, let p if p < q and every n-th splitting node of q is an n-th splitting node of p.
    A fusion sequence is a sequence of conditions {pn : n e u} such that pn pn+1- It is clear that if {pn : n 6 w} is a fusion sequence then n„etJpn is a Cm¿n-tree.
    If s is a node in p, let p j s denote the tree {t G p : t C s or t D s}. If B
    is a set of incompatible nodes of p and for each s € B, qs is a Cmj„-tree such that
    Qs Q P t s , then the amalgamation of {qs : s 6 B} into p is the tree
    {f G p : if t 5 s for some s € B then t G
    12
    Now we start the proof of the theorem by constructing a fusion sequence {pn : n G w}
    of Cmin-trees by induction on the level of the split nodes n: We may assume that
    A = U¿ewA, where each A is closed nowhere dense.
    Let d be large enough such that 2d > m. We will begin our induction stage 0 on
    the set of d-th branching nodes {s0, ■ s2<¿_i} of T. For each s¿ find two nodes
    and s^l^t} such that each |t°| is even and |ij| is odd and for any n-tuple (xo,
    in [T]m with each Xj in a different Cm¿n tree T \ where a, b G {0,1}, we have
    that (xo, ...,xm_i) ^ A)- This can be done using the fact that A0 is closed nowhere
    dense. Let P0 be the amalgamation of {T \ : o, 6 = 0,1}. It is clear that P0 is
    Cmin. Suppose now we have obtained P0,...,Pn at stage n. At stage n + 1 repeat the
    process on the d + n + 1-th branching nodes of Pn such that all the m-tuples selected
    as above from different Cm¿„ subtrees of Pn avoid the set {A0,An+i} and obtain
    T’n+l-
    Let P — CineuPn. Now we have that [P]m fl A = 0.
    â–¡
    Proof of theorem 3.2.2.: By “ remove node s from T" we mean that T is
    to be replaced by its largest subtree not containing s; “ kill the branching node s of
    T" means to remove( in the sense just explained) one of the two immediate successors
    of s. The choice of which successor to remove is arbitrary except when a specific node
    t above s is to be “ retained”; then the immediate successor of s that is not below t
    is to be removed.
    We proceed by downwards induction on r, with n fixed. Since each is
    required to be positive, the highest possible value of r is n, and if r = n each n¿ = 1.
    An n-set from T is then just an n-tuple a of paths through T{. Fix such an
    n-set a. Since the partition is open, all n-sets sufficiently close to a lie in the same
    partition class. Thus, for sufficiently long finite initial segments of a¿, the trees
    T¡ — Ti \ Si satisfy the conclusion of the theorem.

    13
    We turn our attention to the nontrivial case r < n and assume that the
    polarized theorem for r + 1. Without loss, we may assume that for any n-set cr the
    lowest-level branching point /(cr) is in T0. In each of the remaining trees 7¿(1 < i <
    r), all of the paths pass through the same node at the level of /( the r-tuple (/( Lemma 3.2.5. There are Cmin T* C so that any shape -< and parity pattern p,
    all the n-sets having the same signature lie in the same partition class.
    Proof: Each tree T* will be obtained as the intersection of an inductively
    defined decreasing sequence of Cmin subtrees of Tl. At each stage n, build Cm)n
    subtrees If of and select some number ln so that any s € 2 T)71 and below level ln will be in the trees T)m for all m > n. In the end T* = nn To begin the induction, set T)° = T), Z°=0.
    Suppose that, at a later stage n, we have obtained Cm¿„ subtree T-1 and some
    number ln < ui. Choose a level l'n so high that in each T"(¡¡ ^ 0) every node s at level
    Z„ has at least two successors ii, i2 that are branching nodes and such that |ii| is odd
    and |i2| is even; this can be done since each T)n is Kill all the branching nodes
    of between level ln and l'n inclusive and let Zn+1 = l'n.
    Next, choose a branching node / of TJ* above level l'n say at l” and kill all the
    branching nodes between l'n and l” — 1 inclusive, retaining /, and let Zn+1 = Z".
    We now seek to make sure that all n-sets with shape -< and parity pattern p
    with / as the first branching node have their partition classes determined by their
    signatures. Enumerate all the possible signatures (/, S\,..., s*) that begin with /;
    there are only finitely many, say m, of them. Consider each such signature in turn.
    Construct Cmin subtrees T"’°,..., T"’771 of each 7)" by a finite induction as follows:
    Suppose we have constructed up to some T)nJ at some stage j, for stage j -1-1,
    consider:

    14
    The n-set cr in Tn,:? with shape -< and parity pattern p and the j 4- 1-th
    signature (/, sj,sr-i) yields, by splitting the first component, an n*-set
    & ({^0,01 ***i ^-0,^—1}j 4**í ^-0,no—1} ? 1 •••) ^V—l)i
    which is in
    Tnj> = í rox,j rrrsu~xá r*-!>,
    where n* is the (r + l)-tuple (<7, n0 — q, ni,..., nr_i). The shape -<* and parity pattern
    p* of cr* is uniquely determined by -< and p, and cr* determines cr. Partition the
    n*-sets in Tnj’* with shape -<* and parity pattern p* by putting two such cr*’s in the
    same piece if and only if the corresponding partition. Since cr is a continuous function of cr*, this is an open partition of n*-sets,
    and we can apply the induction hypothesis to find Cm¿n subtrees of the 7’nj,*’s such
    that all their n*-sets with shape -<* and parity pattern p* lie in the same piece of
    partition class. Prune the trees T^nj correspondingly and call them T^J+1 so that the
    new
    Tn,j+l rro,TnJ+l rri)j»J+l \8r-l
    are the subtrees of the T/l,J,*’s given by the induction hypothesis.
    Let 77l+1 = Dj Now, let T'* = nn subtree. By carefully selecting the /’s, T0 can be arranged to be Cmin as well. We
    may assume this T* is good for each fixed shape and parity pattern by repeating the
    process finitely many times. This completes the proof.
    â–¡
    We now seek to eliminate the dependence of the partition class on the signature
    for the n-sets with fixed shape -< and parity pattern p, by reducing the trees still
    further.

    15
    Without loss, let us assume that po(0) = 0; i.e., the lowest split node of a0 is
    on the even level. To each r-tuple s with s,• e T* we associate a signature s of T*
    as follows. Let s0 be the highest even level branching node C s0 in . (It will not
    matter how we define s if s0 is below the lowest even level branching node of .)
    For ¿ ^ 0, let Sj be the predecessor or the lexicographically first successor of Sj at the
    level of Sq. Clearly, s = (s0, ...srli) is a signature of T* for the n-sets with shape -X
    and parity p.
    We apply the Halpern-Láuchli partition theorem [6] to the partition of T0* x
    ... x Tr*_! = n T* sending two r-tuples to the same partition class determined by the
    last lemma with shape -<, parity pattern p if their signatures are in the same partition
    class. It asserts that there is a natural number h such that one of the partition classes,
    say c, contains (/i, A;)-matrices for all natural numbers k. This means that, for each
    k, there is a sequence x of nodes at level h in T* and there is a sequence A of
    subsets Ai C T* such that
    (1) every successor of X{ at level h + k in T* is below a node in and
    (2) all r-tuples s € n A lie in the partition class c.
    Fix such an h. There are only finitely many tuples x as above, so the same
    x works for arbitrarily large k\ hence we may assume it is for all Ac. Fix such an x.
    The process of pruning the trees T* begins with setting Si — T* f X{. To simplify
    notation, we assume that h = 0; the general case involves adding h to every level
    mentioned in the sequel.
    Now proceed by induction, at each stage n , build Cm,„ subtrees T*’n of T*
    and select some number ln so that any s € 2 level ln will be in the T*'m for all the m > n. In the end T[ — On the induction, set T*'° = T* and Iq = 0.
    Suppose that at a later stage n, we have obtained Cm¿„ subtrees T*'n and some
    number ln < oj. In stage n + 1, choose l'n so large that, in each T*’n(i ^ 0), every

    16
    node s at level ln has at least two branching nodes t0, ii above s and below level l'n
    so that |i0| is even and |ix| is odd. Next, choose an even level branching node / in
    T*’n above level l'n , say at level Z" > l'n. By Halpern-Láuchli, find A, C T*’n so that
    (1) Ai dominates all nodes of level l" in T*’n, and
    (2) riACc.
    By (1), find s0 G A0 above /. Then s0 is above or equal to /, so its level Z* is
    greater or equal to l'n. Kill all the even level branching nodes of T0*’n from level l'n to
    level Z* — 1 inclusive retaining s0. As before, this ensures that s0 is the unique even
    level branching node of T0*between levels ln and /* inclusive. For any i ^ 0 and
    any e A¿, let s be, as before, the predecessor or lexicographically first successor
    of Si at level Z*. Then € Adominates level l'n of T*’n because Z* > l'n
    and (1) holds. Furthermore, by (2) and the definition of the partition for r-tuples
    other than signatures, {«o} x FL^o ^ included in class c of the original partition
    of signatures. For i ± 0, remove all nodes of T*'n that are not comparable with any
    node in £¿, and let the resulting tree be T*'n+1 and let Z„+i = Z* + 1. All nodes at
    level Z* in T*’n+1(i ± 0) belong to 13¿, so all signatures that start with s0 are in class
    c.
    Now let T¡ — nn Tq can be arranged to contain a Cm¿n tree as well by carefully choosing the /’s.
    â–¡

    CHAPTER 4
    CANONICAL PARTITION THEOREM
    4.1 Perfect Tree Case
    In 1930 Ramsey [11] proved his famous pigeon-hole principle for finite sets:
    Theorem 4.1.1. Let n,r be positive integers and let X be a countable infinite set.
    Then for every partition A : [A]" —> {0, ...,r — 1} of the n-element subsets of X
    into r many classes, there exists an infinite subset Y C X such that the restriction
    A [ [Y]n is a constant mapping.
    A new situation arises if partitions into an arbitrary number of classes are
    considered. For this case, Erdos and Rado [3] proved the so-called canonical version
    of Ramsey theorem:
    Theorem 4.1.2. Let n be a positive integer and let X be a countable infinite set,
    which is totally ordered. Then for every mapping A : [X]" —»• X there exists an
    infinite subset Y C X and a subset I C {0, ...,n — 1} such that for all {ao, ...,an_i},
    {b0, ...,bn-1} G [F]n with a0 < ... < a„_i and b0 < ... < 6n_i we have that
    A({a0,..., a„_i}) = A({60,..., 6n_i}) iff {a¿ : i 6 1} = {6¿ : i e I).
    For many other structures like for example arithmetic progressions, parameter-
    words and finite vector spaces, canonical partition results are known.
    H. Lefmann [7] considered perfect-tree version of these theorems in the space
    with the metric
    d(x,y)— iff k = min{n : x(n) ± y(w)}.
    k +1
    and have the following result:
    17

    18
    For any A G [T]", A = {a0,an_i} and J C {1,n — 1}, let D(A) : J =
    {d(aj-i,aj) : j G J}, where d(x,y) is the distance between x,y.
    Theorem 4.1.3. Let n be a positive integer and let ({l,...,n — l},x) be a total
    order. Further let T Q 2U be a perfect subset and let X be a metric space. Then
    for every Baire-mapping A : [T]" —> X there exists a perfect subset V C T and
    subsets I C {0,n — 1} and J C {1,n — 1} such that for every A, B G [T3]" with
    A — {ttoi '"i ®n—i}<¿ex and B — •••? fin—i}, A(A) = A(B) iff {a¿ : i G 1} = {/?¿ : i G 1} and {d(atj-i,atj) : j G J} =
    {dWj-uPj) :j e J}.
    4.2 Cmir, Tree Case
    Lemma 4.2.1. For any Borel equivalence relation ~ on a Cm¿„ tree T C 2W, there
    is a Cmin tree V C T so that either x ~ y for all x,y G V or x ~ y iff x = y for all
    x,yeV.
    Proof:
    Let ~ be a Borel relation on T. This induces a mapping A* : [T]2 —> {0,1} by
    A*({a0,cn}) = 0 iff a0 ~ ai. Clearly, A* is a Borel mapping. By the Cmin version
    of Blass Theorem, there exists a subtree V C T, either A*([P]2) = i for some
    i = 0,1, or A*(a0, au) = 0 iff a0, cti split on even (odd) level. We have the following
    four cases:
    (1). If this V is such that A*([73]2) = 0, then obviously we must have that
    «o ~ «i for any a¡o, «i G V.
    (2). It is the case that A’QP]2) = 1, then we have that a0 ~ ot\ iff a0 = ct\.
    (3). It is the case that A*(a¡o,Q:i) = 0 iff a0, ai split at an even level. This
    does not happen, since ~ is a Borel equivalence relation.
    (4). It is the case that A*(ci0) ai) = 0 iff a0, au split at an odd level. Again,
    this does not happen since ~ is a Borel equivalence relation.

    19
    â–¡
    Let A C 2U be finite with A ={a0, • ••, an-i} ordered lexicographically. Let
    D(A) ={d(ai-i, a¿) : 1 < i < n} be the set of distances between consecutive elements
    of A. Further let I C {0,n — 1} and JC{1,n — 1}. Then
    A : I = {ai : i G /}
    is the /-subset of A and
    D{A) : J = {<¿(0,-1, a,) : j G J}
    is the J-subset of D{A).
    Theorem 4.2.2. Let -< be a total order on {0,n — 1}, p G 2n_1 and T be a Cmin
    tree. Then for any Borel equivalence relation ~ on \T\\p, there is an I C {0,n— 1}
    and J C {1, —1} and a Cmin V C T such that for any x,y G [P]\p, the following
    are equivalent:
    (1) x ~ y.
    (2) x : I — y : I and D(x) : J — D(y) : J.
    Proof: For n = 1, this is just the lemma.
    For the induction step, use the special constructions from [7]. Let T G
    with T — {ao> an .. Define a mapping / : {0,..., n — 1} —> {0,..., n — 1}
    where f(i) is the unique i+j with j G { — 1,1} such that ajPldi-j is an initial segment
    of o¡i n .
    Definition of T + i.
    For i < n let T + i G [7]n+1 result from T by adding a new element af G T
    to T such that the following is valid:
    (i) af (ii) PI af is an initial segment of Oj fl a¡(¿).

    20
    (iii) for j - 0,1 it holds that if («¡0^) 0 j is an initial segment of af; thus
    af leaves atf^ in the same direction as does.
    (iv) T and (T \ {a¿}) U {a+} have the same shape and parity pattern.
    Definition of T © i.
    For i < n let T © i £ [T]n+1 result from T by adding a new element af £ T
    in the following way:
    (i) of (ii) Oj fl a/(j) is a proper initial segment of a¿ ft af
    (iii) T and (T \ {a,}) U {a®} have the same shape and parity pattern.
    Let the resulting sets T + i resp. T © i have shape - resp. -<® and pf. T + i or T © i need not exist for every T € [7”]")P, but certainly
    there exist subsets T £ [7"]" tP such that T + i and T © i exists. If they exists, they
    are in general not unique, while their order-types, and only these will be of interest
    in the following, are independent of the special choice of the added elements.
    Now we start with the induction step. Let n > 2 be given and assume that the
    theorem is valid for all k < n. Let ~ be a Borel equivalence relation on [T]" >p. For
    every i < n the equivalence relation ~ induces a mapping : [T]"^1 + —> {0,1}
    “N >Pi
    by
    At(T) = o iff T : ({0,..., n} \ {i}) ~ T : ({0,..., n}\{i + 1}).
    Since the functions Af are Borel mappings, by Theorem 3.2.2., there exists a
    Cmin subset % C T and for every i < n there exist a number c £ {0,1} such that
    A.nrr„]^« > = c
    Case 1: If this % is such that A+([7o]n+1 +) = 0 for some i.
    ~ For T £ [7o]" let the resulting set T : ({0,..., n — 1} \ {¿}) have shape -<* and
    parity pattern p*. Then ~ induces a equivalence relation on [7o]"Vp. by
    (T0 : ({0,..., n - 1} \ {*})) (Tx : ({0,..., n - 1} \ {*})) iff T0 ~ Tx.

    21
    Restricting to a Cmin subtree of To if necessary, we may assume that is Borel and
    well-defined.
    By induction hypothesis there exists a Cm¿n subset Tx C % such that there is
    some T C {0,n — 2} and J* C {1,n — 2} such that
    A B iff A : I* = B : V and D(A) : J* = D(B) : J* for any A, Be
    Now for any T0, Tx e [7”]")P, let I C {0,..., n — 1} and J C {1,n — 1} be such that
    for T e [7í]")P, we have that
    (T:({0 n—1}\M)):/-=T:/
    and
    D(T : ({0,n - 1} \ {z})) : J* = D(T) : J.
    It follows for T0,Ti e [71]" tP that
    T0 ~ Ti iff T0 :1 = Tx : I and D(T0) : J = D(Ti) : J.
    Case 2: If To is such that c = 1 for all i:
    For i < n the equivalence relation ~ induces a mapping A® : [73]"®^® ~^
    {0,1} by
    Af (T) = 0 iff (T : ({0,n}) \ {*})) ~ (T : ({0,n} \ + 1})).
    We may assume that each A® is continuous. By Theorem 3.2.2., there exists
    a Cmin subtree T\ C To and for each i < n and dj e {0,1} such that
    aftmiift.) = *•
    Consider for every two-element set {7o,Ti} € [[7i]" p]2 the union To U 7i, say \Tq U
    7j| — p, of shape -<* and parity pattern p*, let K0, Kx C {0, 1} be subsets such
    that
    (T0 U Ti) : K0 = T0 and (T0 U Tx) : Kx = Tx.

    22
    Let (({1, ...,Pi}, - ring triples. For every i < q let A¿ : [Ti]% p. —> 2 be a mapping defined by
    Ai(T) = 0 iff (T : Kf) ~ (T : K¡).
    Since the functions A¿ are continuous, by Theorem 3.2.2., we have a
    subtree T2 C 71 such that
    Ai \ [T2]% = Cj,where c¿ G {0,1} for every i.
    For any T G let / = {i < n : d{ — 1}. We claim that for all T0,Ti G
    [T2]”iP, we have that
    To ~ Ti iff T0 : / = Ti : I and £>(T0) = D(Ti).
    For i — 0,1, let = {oq, ..., aln_ 1} be given. At first we show the implication
    from right to left. So let T0 : I = 7\ : I and D(T0) — D{T\). If I = {0, ..,n — 1},
    there is nothing to prove. So let I ± {0,..., n — 1} and take l G {0,..., n — 1} \ I.
    Since d¡ — 0, we conclude that
    {a°,..., a°_1} Iterating this we get
    {ag,..., Q^.iKíex ~ {a,0 : i G lo} U {a) : j G {0,..., n - 1} \ 1}
    and thus T0 ~ T\.
    Now we prove the implication from left to right. Let T0 ~ Tj and assume
    T0 : / ¿ Ti : I or D(T0) ± D{TX). Suppose D{T0) ± D{TX). Let
    d = mm((D(T0) \ D(Ti)) U (D(TX) \ D(T0))),
    where without loss, d — o?(a°_1, ot¿). Since d is minimal, there is j < 2 such that
    a — (a¿_1 fl af) ^ j is not an initial segment for any ¡3 G T\.

    23
    Let a be an initial segment of, say, aChoose a¿ G T¿ \ (To U T\) such that
    (i) (a^ fl ái) ^ j is an initial segment of a¿.
    (ii) a is not an initial segment of a8.
    (iii) T0 U Ti and ((T0 U T{) \ {a-1}) U {á¿} have the same shape and parity
    pattern.
    Choosing other sets T0,Ti if necessary, this is possible. But then we have
    T0 ~ Ti ~ (T0 \ {a°}) U {ái},
    contradicting the fact that A/'([7<)]"+1 +) = 1 for every i.
    'i 'Pi
    Suppose now D(T0) — D(T\) but T0 : I ^ T\ : I. Then there exists i € I with
    a? ^ aj, say a£ T0 U Ti and ((T0 U Tx) \ {aj}) U {a} have the same shape and parity pattern. As
    before, we get
    T0 ~ Tx ~ (Tx \ {a,1}) U {a},
    which contradicts the fact that = 1. This completes the proof of the theorem.
    â–¡

    CHAPTER 5
    HALPERN-LÁUCHLI THEOREM
    5.1 Introduction
    In the following we will describe some of the the known facts about the
    Halpern-Lauchli Theorem. We will prove a Cmin version of the Halpern-Lauchli
    Theorem and an infinite version of the Cm¿n version of the theorem. Please refer to
    Chapter 2 for basic definitions.
    Theorem 5.1.1. ( Halpern-Lauchli) Let d be a natural number, let be a perfect
    tree of height uj for every i < d, T= (T¿ : i < d), and let k be a natural number. For
    every function f : 0 T—> k there is an A £ [w]w and a perfect subtree U{ C 7¿ for
    every i < d such that the function f f is constant, or in other words, ®AXJ
    is homogeneous for f.
    Let us call the above theorem HL¿. In [8], the infinite version of the Halpern-
    Lauchli theorem HLU was proved. It has two other equivalent restatements.
    Theorem 5.1.2. The following are equivalent:
    (i) HLU
    (ii) If VK is the partial ordering for adding k side-by-side Sacks reals with
    countable supports, then every X C uj in Vv* contains, or is disjoint from, some
    Y e [w]w with Y 6 V.
    (iii) If fi â–  i < uj are continuous functions from the Hilbert cube [0, l]w into
    [0,1], then there exists non-empty perfect sets Pi C [0,1](¿ < uj) and A E [uj]u such
    that, on 0i 24

    25
    5.2 Cmin Tree Case
    Theorem 5.2.1. ( The Cmin version of the Halpem-Lauchli theorem) Let d,k be
    natural numbers, {T¿ : i < d} be Cmin trees. Then for any function f : (&Ti —> k
    there is an A € [ai]u and Cmin subtrees U, C such that f f (^)'4 U= c for some
    c < k.
    The theorem can be easily proved by induction using the following [13]:
    Definition 5.2.2. Let T be a tree. A set ACT is a n-dense set if and only if there
    is an m > n such that A C T(m), the m-th level ofT, and for every s G T(n) there
    exists at G A such that IDs. Similarly, ifT0,..., Td-X are trees and Ai C T¿(m) for
    all i = 0,..., d — 1, then the sequence (An ■ Ad-1) is n-dense set in (T0,..., T¿-\) if
    and only if for every i < d the set At is n-dense in Ti.
    Theorem 5.2.3. (Dense set version of Halpem-Lauchli Theorem)
    Let d, k be natural numbers, {T, : i < d} be a set of perfect trees. Then for any
    function / : 0 T¿ —> k, there is an A G [u;]", such that either one of the following
    is true:
    (i) Vn £ uj, there is an n-dense set (^40j —5 A¿-i) in (To,...,Td_i) such that
    f"A0 x ... x Ad-i = {0}, or
    (ii) There is a t E T0 x ... x Td-\ such that for all n G u there is an n-dense
    set (B0,...,Bd-i) in (T0[t0], ...Td-i[td-i]) such that f"B0 x ... x Bd-X = {1}.
    Proof of Cmin-Halpern-Lauchli Theorem: We may work on one dimension
    only, because for finite dimension products, the proof is similar. Given any Cm¿„ tree
    T, without loss of generality, let us assume that the first case of the dense set version
    of the Halpern-Lauchli Theorem occurs. First let n — 0, and find no and A0 that
    satisfies the dense set theorem, trim all the nodes that are not comparable with any
    node in A0, i.e cut all the nodes s with no r € A0 such that r C s or r D s. Now find
    an m so large that every node t in A0 there are at least two branching nodes íi,¿2

    26
    above t such that |f0| is even and |¿i| is odd. Now find n\ > m and A\ using Theorem
    5.2.3.. Repeat the process infinitely many times, we easily get a tree V that contains
    a Cmin-tree. Eventually, the set {n¿ : i < w} is the set A required in the Cmin version
    of the Halpern-Láuchli Theorem.
    â–¡
    Theorem 5.2.4. HLU is true for the Cmin trees.
    Proof: If d < u and m < k < u, X = (Xt : i < d), X C ®i some trees T¿, let Am= {(: i < d) : for each i, C Xi, CardR¿ = 4 and each
    Bi = sh t^to s2, t^tfsi} for some t G T¿(m) and |i0| is odd and |ii| is
    even }.
    Call S a 4-branch subtree of the Cm¿„ tree T if S is the union of four distinct
    maximal branches in T and such that and where \to\ is even and |ii| is odd,
    are the branching nodes above the stem (of the four branches) t.
    Lemma 5.2.5. Let d < u and T¿(z < d) be Cmin trees. Let m < uj. Then there
    is p(m,T— (Ti : i < d)) < uj, large enough so that for any G C ®i Gfl(^)B^0 for all Be Ajn((Ti(p) : i < d)), then there is an m-dense (Yi : i < d),
    Yi C Ti(p), with 0 YC G.
    Proof: This is a consequence of the dense set version of the Halpern-Láuchli
    Theorem. If for some m and all p a counterexample GPim existed, then UpGp be a subset of 0w(TÍ : i < d) not containing the product of an m-dense sequence; by
    the dense set version of the Halpern-Láuchli Theorem, there exists t G 0T and for
    infinitely many p, a t-p-dense sequence Xp with 0 Xp nGp,m = 0. Choose p such
    that each G T, has two splitting nodes V'to and 1Tt\ above it such that |i0| is even
    and |ii| is odd and the splitting nodes are below p; this is a contradiction.
    â–¡

    27
    It suffices for the general case to prove the Cmjn version of HLw for those
    sequences (T¿ : i < uj) of Cm¿„ trees such that
    limi Let be the collection of such sequences. For T £ 7^, an n-dense sequence
    in T is an (X¿ : i < uj) such that for some m, each Xi C T¿(m), (Xo,^n-i) is
    n-dense in (T0, ...,Tn-i), and X¿ / 0 for i > n. If x £ 0T, an x-n-dense sequence
    is an n-dense sequence in (7* (: i < uj).
    Lemma 5.2.6. If T £ Tu, A £ [w]K°, G C 0^ T and for each n < uj there is an
    n-dense X with 0X CG, then there exists Cmin Ti C Ti(i < w) and A! £ [^4]H° with
    0A' T' CG.
    Proof: This is clear.
    â–¡
    Lemma 5.2.7. If T £ Tw, A € [cj]n° and 0'4 T= Go U Gi then either
    (a) for all m there is an m-dense X with 0X C Go, or
    (b) for some non-zero d < uj there exists 4-branch subtrees Si of Ti(i < d),
    Cmin subtrees T[ ofTi(i > d), A! £ [v4]K°, such that
    (*) for any infinite set A" £ [^4']Ko and any sequence (Tf : i > d), with T" a
    Cmin subtree of T[, there exists t £ 0'4 T", say height t= q, such that for any u
    € ®i € Gi.
    Proof: Suppose that (a) fails for m, then m will be the d of (b). Let p be
    the p(d(Ti : i < d)) of lemma 5.2.5.. Thin down each T¿ to T¿(¿ < d), such that is
    the same as T¿ at levels < p, and above level p consists of, for each node t £ T¿(p),
    a single maximal branch of T¿ above t. Let a0,...,ai enumerate Am((Ti(p) : i < d)).
    For each p' > p this induces an enumeration ao(p'),...,ai(p') of Am((Ti(p') : i < d)).

    28
    Now construct a sequence by induction on /' < l as far as the construction
    can be continued,
    V? D V> D ...Vf
    of downwards closed Cmin subtrees of Tu for each i > d, and a sequence
    B° D B1 D ... D B1' D ...,
    with each BJ an infinite subset of A, so that for each /' < /, for any tG 0 {V* :
    i > d) there exists u G 0ai< (height t) with u~t G G0.
    This thinning process cannot be continued through all / 4- 1 stages. Otherwise
    there would be tG 04(T¿ : i < of), level t > p such that for all l' < l there exists
    uG 0ai' ( height t) with u~t G Go- For u G 0¿ if u~t G Go- By definition of p, Go would contain an m-dense sequence; appending
    t to it would give an m-dense X in (T¿ : i < u) satisfying (a).
    Thus we shall get to a l' < l where Vf, B[' can be chosen but not Vf+1, B\ +1.
    Let Al = B1', T[ — Vf (i > d) and let 5¿(/ < d) be the 4-branch subtrees of T¿(¿ < of)
    determined by ai<+i. These satisfy the property (*).
    â–¡
    Lemma 5.2.8. 7/T G % and 0U'T= Go U Gi, then either
    (a) for all m there is an m-dense X in T with 0X C Go, or
    (b) there are 4-branch subtrees Si ofTi(i < u) and A G [w]N° with 0'4 S C Gi.
    Proof: By Lemma 5.2.7., pick a non-zero d0 < w, A0 G [w]No, 4-branch
    Si C Ti (i < d0) and Cmin T® C T¿(/ > of0) such that for any B G [^4]N° and Cmin
    Ui Q Ti(i > d0), ®U(Ui : i > do) O Go # 0, where we define, for t G04°(í7í : i > of0),
    t G G? if and only if for every u G 0^ Also, arrange that there exists tG 0^(7)° : / > d0), tG G°, such that no
    Tj° splits before height (t) by thinning down the If if necessary; let height (t)= a0,

    29
    the first member of A. By the assumption that (a) fails, there is m such that no
    m-dense sequence X in (If : i > d0), with height X G j40, satisfies (^)XnG° ^ 0.
    Thus we may carry out the construction again with respect to (Tf : i > do), Aq
    and G°; that is, using lemma 5.2.7, find 4-branch S¿ C Tf(do < * < di), Cm¿„ tree
    Tf C Tf(i > di), A! G [^4o]N°7 and Gj(tG G\ if for every uG ®d0 level as t, iGtG G?), with (U{ : i > d\) fl G} ^ 0 whenever each Ui is Cm¿„ in Tf
    and B G [x4']N°, and thin out to pick ai > o0 as before. Repeat this construction for
    infinitely many times, getting (Si : i < u) and A — {an : n < a;}; these satisfy the
    lemma.
    â–¡
    To prove the theorem, we are given T= G0 U G1? and will construct Cm¿n
    trees Ui(i < lj) and a map from U into 0U'T, and show that a homogeneous
    (g)'4 S (Sj a 4-branch subtree of Ui) induces a homogeneous 0s T' (T' a Gmjn subtree
    of Ti).
    We may assume that for each i,
    Ti = {se 2<“ : s(j) = 0 Vj < *} = Ki,
    for (Ki : i < u) is isomorphic to a sequence of subtrees of (T¿ : i < u) restricted to
    a common infinite set of levels. Let (Bn : n G uj) be a disjoint partition of uj, with
    each Bn infinite. We may assume that each Bn has infinitely many odd numbers and
    even numbers in it. Now, via some bijection between uj and u x w, {Uj : j < cj} will
    be {Ui Gi,„ := {s G 2 Then, if u G®WU, height u =m, define H(u) G(^>^T, height t —m by
    the rule:
    Ui,n if i < j < m, j e Bn,
    0 if j < i.
    U(j) =

    30
    Then H is a level preserving bijection between 0T U and 0 T. We have now
    0WU= Go U Gi, where Gj H~1”Gj. By the last lemma, there exist an infinite
    set A C u) and 4-branch subtrees S¿iTl of UitTl(i,n < uj) and j < 2 with 04 SC Gj.
    Then there are downwards closed subtrees T{ of Tj with H" 0^ T', and we have
    (g^T'C Gj. It is clear that each is Cminâ–  This completes the proof. â–¡
    Definition 5.2.9. A nonprincipal ultrafilter U on u is selective if for every sequence
    (An : n < u) of elements ofU there is A eU such that A \ n C An for every n € A.
    Such A is said to be the diagonalization of the sequence (An : n < uj).
    Lemma 5.2.10. A non principal ultrafilter U on u is selective iff for every partition
    p : [w]2 —> 2 there exists i < 2 and A eW such that p''[A]2 — {*}.
    Lemma 5.2.11. IfU is a selective ultrafilter on üj and if A is an analytic subset of
    [ufff, then there is an A Ell such that [A]u C A or [,4]“' fl A = 0.
    Definition 5.2.12. IfU is an ultrafilter on uj, let HLd(U) be HLd with the require¬
    ment that the set A be a member ofU.
    Theorem 5.2.13. For every d < u and every selective ultrafilter U on ui, the Cmin
    version of HLd(U) is true.
    Proof: Let Tfii < d) be a given sequence of Cm¿n-trees and let p :
    {0,1} be a given coloring. If d = u, we further assume, without loss of generality,
    that if Si is the minimal splitting node of for i < uj, then the length of increases
    with i. Let
    A = {Ae : (V* < d)(3 Cmin Ui C 7¿)(3e < 2)p" ® U \ A = {e}}.
    i Then A is clearly an analytic subset of [w]w so by fact 2, there exists A G U
    such that either [A]“ C A or [i4]w fl A — 0. Note that by the Cmin version of
    Halpern-Láuchli theorem, the second alternative does not happen, so we are done.
    â–¡

    31
    Theorem 5.2.14. Any finite product of Cmin forcing preserves the selective ultrafil¬
    ter, hence does not add any independent sets.
    This follows from the next theorem.
    Theorem 5.2.15. Let P be the Cmin forcing. Every selective ultrafilter U generates
    a selective ultrafilter U* in the forcing extension VPu.
    Proof: The only thing left to prove is that
    U* = {A c u : 3B e U such that B C A} is selective;
    that is, every sequence (An : n < u) of elements of U* has a diagonalization in U*.
    But this is easy to show. Consider a fusion sequence P‘, where s € 2, such
    that for every n < u and s= (s, : i < u) in (2n)w the condition (P*. : i < u) forces
    that An is equal to some element Bs of U, we may do so since we can without loss
    of generality assume that each An is taken from U. Let (P¿ : i < u) be the fusion of
    this sequence and let B G U be such that B \ n C Bs for every n G B and s€ (2")“.
    This finishes the proof.
    â–¡
    We also have the analogue of the perfect set result on Hilbert cube:
    Theorem 5.2.16. For every sequence {/„ : n < u>} of continuous functions from the
    Hilbert cube [0, l]w into the interval [0,1], there is a subsequence {/¿} and a sequence
    {P„} of Cmin subsets of[0,1] such that {/¿} monotonically converges to a continuous
    function on the product P0 x Pi x P2....
    Proof: Let P be the Cmin forcing. In the forcing extension of P“ look at the
    sequence of reals /n((s¿ : i < tu)), where (s,) is the canonical name for the generic
    sequence of Cmin reals. Under the assumption that there is a selective ultrafilter U,
    U* is a selective ultrafilter, apply the above fact to the partition, p : [w]2 -» 2 defined

    32
    by
    p({m,n}) = 0 iff fn((¿i : i < w)) < ■ i < w>),
    and get a member A of U* such that /„((s¿ : i < cu)) (n G A) is monotonic. Fix
    (Pi : i < u) e P" deciding the set A and the fact that the sequence is, say, increasing.
    Let
    F - {(xi : i < u) e Y\Pt ■ (Vm < n G A)fm((xi : i < u)) < /„((x¿ : i < w))}.
    i Notice that F is a closed subset of n¿ of this product contains a product of a sequence (P{ : i < u) of Cm,n subsets which
    must also force that the sequence /n((s¿ : i < u)) is increasing, the set F must be
    equal to the product Let 9 â–  Yli Since g is Baire, by further restricting to the subtrees if necessary, we may assume it
    is continuous. So we are done.
    â–¡
    Theorem 5.2.17. For every finite d and a sequence {/„} of continuous functions
    from the cube [0, l]d into [0,1], there is a single Cmin set P C [0,1] and a subsequence
    {/¿} which is monotonically convergent on Pd.
    Proof: The set P will be the result of a fusion sequence Ps(s G 2 structed together with a decreasing sequence of set An(n < u) such that for every
    n < u and every sequence of distinct elements of 2n the subsequence {/*, : k G An]
    when restricted to the product [li function. Clearly, there is no problem in constructing this fusion using the previous
    result. It should be pointed out that if {kn}n that kn G An for every n < u then {F1A:n}7i the theorem.
    â–¡

    CHAPTER 6
    OTHER FORCING NOTIONS
    6.1 The En Trees
    Let E0 be the relation on M, such that
    for any x, y E K, x ~ y iff x and y differ by a rational.
    It is easy to see that E0 is indeed an equivalence relation. When working with
    the space 2U, let x ~ y mean that x{n) — y(n) for all but finitely many n.
    Let I be the ideal cr-generated by the partial Borel Eo-selectors, where a partial
    selector is a subset of M which meets each equivalence class in exactly one point. It
    is well-known that Borel E0 selectors are meager and it is easy to show that any
    member in the E0 ideal / has Lebesgue measure 0, hence Eq is nontrivial. For more
    information about the E0 ideal, see [14].
    Definition 6.1.1. An E0-tree is a non empty tree TC 2“ which is perfect, that is,
    every node in T can be extended into a split node, and homogeneous, i.e., for every
    split node t e T there are two longer split nodes s0, si of equal length such that every
    node compatible with t is compatible with Sq or s\ and {u : s^u € T} — {u : sfu G T}
    Theorem 6.1.2. [If.] Every analytic subset of 2W is either in the ideal I or it contains
    all branches of some E0-tree.
    The following result is due to Blass in a private conversation:
    Theorem 6.1.3. There is a Borel function f : [2W]3 —> 2U such that for any E0 tree
    S C 2W, we have that /”[5]3 = .
    33

    34
    Proof: Define a Borel function / from [T]3 to 2W, where T is any E'o-tree,
    such that for any Eq subtree S C T we have that /([S']3) = 2W. Given any set
    {xi,x2,x3} ordered lexicographically, we may assume that all of the pairs {xi,xj)
    differ at infinitely many places for all 1 < i < j < 3. Let
    m(xi,x2) = 3, m(x2,x3) = 1, m(xi,x3) - 2.
    Since we are working on 2U, there must be a least natural number k0 and a
    pair (x°,Xj) selected from the set {xi,x2,x3} such that x®(k0) — x°j(k0)', for n > 1 let
    the natural number kn be the n-th smallest number such that for some pair (x]\x"),
    we have that x™{kn) — x](kn) and such that the pair (rr",x") ^ Define
    f(xi,x2,x3)(n) to be |m(x",x]J) — m(z",a:")| mod 2, where (x",x") are such
    that xnj(kn) = x%(kn) and x?(kn+1) = x^(kn+1) â–¡
    Observe that if we divide IE into m disjoint open balls, we get a function that
    will hit every partition class for any E0 tree S C 2“. Hence no partition theorem is
    possible.
    6.2 The Silver Forcing
    In the following we study the Silver forcing (See [14] for more detail). The
    Silver ideal is a variation of the £o-ideab Let G be the graph on 2W connecting two
    binary sequences if and only if they differ in exactly one place. Let I be the ideal
    ^-generated by Borel G-independent sets.
    Definition 6.2.1. A Silver tree is a nonempty tree T C 2“ which is perfect, and
    homogeneous in the sense that for every split node t G T every node compatible with
    t is compatible with T'O or V' 1, and {u : G'O^u E T} — {u : t^l^u € T}.
    Theorem 6.2.2. [If] Every analytic subset of 2“ is either in the ideal I or it contains
    all branches of some Silver tree.

    35
    The following results were generalized by Blass in a private conversation after
    I told him how to construct a continuous function / : [2W]2 —>• 2 such that for any
    Silver tree S we have that /”[5]2 = 2.
    Theorem 6.2.3. There is a continuous function f : \T]n —> m, where T is any Silver
    tree, such that for any Silver tree S C T we have that /” [S]n = m.
    Proof: Define a continuous function / from [T]n to n, where T is a Silver
    tree, the following way:
    Given a Silver tree T and {x¡ : i < n} set of elements of T, let s be the lowest
    branching point for the set {x¿ : i < n}, and let k be the number of l’s in s modulo
    m. Define /({x¿ : i < n}) = k.
    It is easy to check that no homogeneous Silver subtree 5 C T for / is possible.
    â–¡
    Theorem 6.2.4. There is a continuous function f : [T]n —> uj, where T is any Silver
    tree, such that for any Silver tree S C T we have that /”[S]n = u.
    Proof: Given any Silver tree T and n-set {x¿ : i < n}, let s be the lowest
    branching point of the n-set as before, now let k be the largest number m such that
    2m divides the number of l’s in s.
    It is easy to check that this definition works.
    â–¡
    Theorem 6.2.5. There is a Borel function f : [T]2 —> 2W, where T is any Silver tree,
    such that for any Silver subtree S C T we have that /”[S]2 — 2W.
    Proof: Given any X\,X2 E T, without loss we may assume that X\,X2 differ
    at infinitely many places. Define the function / by induction: At the n-th stage look
    at the n-th point that X\,X2 differ, if it is the case that x\(n) = 0, X2(n) = 1 then let
    f(xi, x2)(n) = 0, otherwise, f(x\, X2){n) — 1. If xi,x2 differ by only finitely many,
    say, n places, let f(xi,x2)(m) — 0, whenever m > n.

    36
    â–¡
    6.3 The Packing Measures
    In the following we define the packing measure on the space 2W with the metric
    d(x,y) — 2n+ii where n is the least number such that x(n) ± y(n). For any positive
    real number h < 1, any set A C 2W, we say that {(x^r*) : Xi G A} is a 5-packing on
    A if the open balls containing with radius r, are disjoint and 2< 5 for any i. Let
    P£s(A) =sitp{^rf : {(x¿, r¿) : i < w} forms a 5-packing }.
    let
    Pq(A) = limS-K>Po,g(A),
    and define the packing h-measure for A C 2W by
    OO
    Ph(A) = infiY'PoM â–  A C
    i=1
    Let I/t be the a -ideal generated by the subsets of IP with finite packing h-
    measure.
    Theorem 6.3.1. For any perfect tree T, there is a continuous function f : [T]3 —> 2,
    such that for any I^-positive S CT, we have that /”[S]3 = 2.
    Proof: Fix a sequence of partitions (/„ : n < u) such that each fn : [n]2 —> 2
    and each fn witness the smallest possible size of homogeneous set. By a combinatorial
    theorem [2], which stated that
    rk (2k - 2\ ,
    ^ where n* is the least natural number such that for any partition / : [n*]2 —» 2
    there exists a homogeneous subset of size k, let dn be the smallest number such that
    y/2dn > n. Given any {x0, X\, x2} G [T]3, we will get exactly 2 forks, where a fork is an
    s G 2
    37
    Go to the level of the tree where the highest of the 2 forks are located, say, level n.
    We have exactly 2 nodes, say s,t € 2" that are initial segments of x0,x\ or x2. Now
    look at all the nodes at level n and let us say that we have totally m elements on
    the n-th level of the tree ordered lexicographically and say that s is the ¿-th element
    and t is the j-th element of the n-th level of the tree. Let /(x0, £i, x2) =
    Now it is easy to see that under this definition / is really continuous. We can easily
    compute the weight at the n-th level, which is
    D^>"+1)‘ = m • ((¿)”+1)'‘=<*■
    The biggest size of homogeneous set that we are able to obtain is less than dn. Now
    the size of the homogeneous set in the next level is at most dn + 2 by the last theorem.
    So the weight on the next level is < (dn + 2) • a • (|)fc. So the sum of all the weights
    at all the levels is easily seen to converge to a finite value using the ratio test. Hence
    it is impossible to have a /^-positive subset that is homogeneous.
    â–¡

    REFERENCES
    [1] A. Blass, A partition theorem for perfect sets, Proc. Amer. Soc., 82 (1981),
    pp. 271-277.
    [2] M. Bona, A walk through combinatorics, World Scientific, Hackensack, 2002.
    [3] P. Erdos and R. Rado, A combinatorial theorem, J. London. Math. Soc.,
    25 (1950), pp. 249-255.
    [4] S. Geschke, M. Kojman, W. Kubis, and R. Schipperus, Convex decom¬
    positions in the plane and continuous pair coloring of the irrationals, Israel J.
    Math., 131 (2002), pp. 285-317.
    [5] F. Galvin and S. Shelah, Some counterexamples in the partition calculus,
    J. Combinatorial Theory Ser., A15 (1973), pp. 167-174.
    [6] J. D. Halpern and H. Lauchli, A partition Theorem, Trans. Amer. Math.
    Soc., 124 (1966), pp. 360-367.
    [7] H. Lefmann, Canonical partition behaviour of cantor spaces, in Irregularities
    of Partitions, edited by G. Halász and V.T. Sós, Springer-Verlag, New York,
    1989, 93-105.
    [8] R. Laver, Products of infinitely many perfect trees, J. London Math. Soc., 29
    (1984), pp. 385-396.
    [9] J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1989),
    pp. 139-147.
    [10] J. Mycielski, Algebraic independence and measure, Fund. Math., 61 (1967),
    pp. 165-169.
    [11] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 30
    (1930), pp. 264-286.
    [12] S. Shelah, and J. Zapletal, Duality and the pcf theory, Mathematical
    Research Letters, 9 (2002), pp. 585-595.
    [13] S. Todorchevich and I. Farah, Some application of the method of forcing,
    Yenisei, Moscow, 1995.
    [14] J. Zapletal Descriptive set theory and definable forcing, AMS Memoirs, Pxox-
    idence, 2004.
    38

    BIOGRAPHICAL SKETCH
    Yuan-Chyuan was born in Taiwan in 1973. He spent the first 25 years of his
    life on that island. He came to Gainesville to study mathematics in 1998. In 2001
    he began studying under Professor Jindrich Zapletal and working toward a doctoral
    degree.
    39

    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Dbctor of Philosophy. /] /)
    ll.u
    Jindrich Zapletal (phairman
    Assistant Professor of Mathematics
    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of Philosophy.
    Jean A. Larson
    Professor of Mathematics
    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor
    Wifham Mite
    Professor of Mathematics
    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor of Philosophy.
    Rick L. Smith
    Associate Professor of Mathematics
    I certify that I have read this study and that in my opinion it conforms to
    acceptable standards of scholarly presentation and is fully adequate, in scope and
    quality, as a dissertation for the degree of Doctor oLPhilosopdw.
    LJL
    Greg Ray ' /
    Associate Professor of Philosophy
    This dissertation was submitted to the Graduate Faculty of the Department of
    Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
    and was accepted as partial fulfillment of the requirements for the degree of Doctor
    of Philosophy.
    May 2005
    Dean, Graduate School

    , vSi>5^
    UNIVERSITY OF FLORIDA
    3 1262 08554 2412



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