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PARTITION PROPERTIES AND HALPERNLAUCHLI THEOREM ON THE C,mi FORCING By YUANCHYUAN 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 HALPERNLAUCHLI 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 HALPERNLAUCHLI THEOREM ON THE Cmn FORCING By YuanChyuan 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 Cminideal I. We define a special perfect subsets of the space 2", called Cmin trees. We study the partition properties and HalpernLiuchli 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 HalpernLuchli 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 FirstOrder 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 HalpernLiuchli 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 aideal 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 Ipositive 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 pigeonhole 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 socalled 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 HalpernLiuchli Theorem. In a 1984 paper, Richard Laver [8] generalized the existing HalpernLiiuchli 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 HalpernLiiuchli 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 wellorders A, or (A, R) is a wellordering iff (A, R) is a total ordering and every nonempty subset of A has an Rleast 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 wellordered 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 nset 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 nelement set {ao, .., an1} C [T], we mean the linear ordering of {1, ..., n 1} given by i < j + d(ai1, ai) is in a lower level than d(aoj_, aj). Definition 2.1.6. By the parity pattern p, of an nelement set {ao, ..., an1} E [T]n, we mean p E 2"' given by p(i) = 0 if the ith (counting from 0) lowest splitnode of the nset 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 ntuples 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 ntuples 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 aalgebra 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 aalgebra S such that S D X; namely the intersection of all aalgebras 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 aalgebra 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 aalgebra; 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 Cminideal be the aideal agenerated 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 nontrivial. 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 onetoone 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,..., Tr1) be a skew rtuple 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 rtuple of positive integers with sum n. By an nset in T we mean an rtuple a whose ith entry oa is an nielement subset of (Ti). The shape < of such an nset 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 jth 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 nsets. Let the collection of all nsets in T be partitioned into finitely many open pieces. Then there exist perfect trees T' C Ti such that all nsets 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 Cmintree has a Cmin skew subtree. Proof: Given any Cmintree 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 nset r in a sequence of trees T is defined by p (i) = 0 iff the ith lowest splitnode in aj is on the even level. Theorem 3.2.2. (Polarized Theorem for the Cmin trees) Let T=< To, T1,...,Tr1 > be a set of Cm,, trees, n=< no,...,n,1 >, and let < be any shape of nsets, p be a parity pattern of nsets. Let the collection of all nsets in T be partitioned into finitely many open pieces. Then there exist Cmin trees Ti C Ti such that all nsets 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 Citree, and let [T]" be partitioned into a finite number of open( in [T]n) pieces. Then there is a Cmintree S C T such that [S]n intersects at most 2"' x (n 1)! of the pieces. Note that the 2n1 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 Cmintree 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 Cmintree 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 Cnintree. 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 nth 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 nth splitting node of q is an nth 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,itree. 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 Cmintree 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 Cmintrees 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 dth 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 ntuple (xo,..., xm1) in [T]m with each xj in a different Cmin tree T [ sa^t, where a, b {0, 1}, we have that (xo, ..., Xm1) 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 + 1th branching nodes of Pn such that all the mtuples 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 nset from T is then just an ntuple a of paths ai through Ti. Fix such an nset a. Since the partition is open, all nsets 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 nset o the lowestlevel 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 rtuple (f(ar), sl,..., sri) the signature of the nset a. Lemma 3.2.5. There are Cmin Ti* C Ti so that any shape < and parity pattern p, all the nsets 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 nsets 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 nset r in T"'J with shape < and parity pattern p and the j + 1th signature (f, sl, ..., Sr) yields, by splitting the first component, an n*set c* = ({ao,..., oq1, o,q, ,, O,nol a1, ', r1), which is in Tn"j, = (To'j [ fr O, T~"j f'l,T'n, r si,...,TrTJ r sr1), where n* is the (r+ 1)tuple (q, no q, n, ..., nri). 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 [ S1 are the subtrees of the T "~'*'s given by the induction hypothesis. Let T+1 = nj 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 nsets 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 rtuple 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 nsets with shape < and parity p. We apply the HalpernLiuchli partition theorem [6] to the partition of T* x ... x Tr = T* sending two rtuples 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 rtuples 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 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 HalpernLiiuchli, 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 rtuples 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 pigeonhole 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 nelement 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 socalled 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,..., bn1}) 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 perfecttree 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(a1, 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 Bairemapping 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,..., aLn1}<,ex and B = {(o, .., /n1} {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, ...,an1} 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 Isubset of A and D(A) : J = {d(ajl, a) : j E J} is the Jsubset of D(A). Theorem 4.2.2. Let < be a total order on {0,..., n 1}, p E 2n1 and T be a Cmin tree. Then for any Borel equivalence relation ~ on [T],,, there is an I C {0,..., n1} and J C {1,..., n1} 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 ainaij 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 