ASPECTS OF STABILITY IN SIMPLE THEORIES
A Dissertation
Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Donald A. Brower
Steven Buechler, Director
Graduate Program in Mathematics Notre Dame, Indiana April 2012 c Copyright by Donald A. Brower 2012 All Rights Reserved ASPECTS OF STABILITY IN SIMPLE THEORIES
Abstract by Donald A. Brower
Simple theories are a strict extension of stable theories for which non-forking independence is a nice independence relation. However, not much is known about how the simple unstable theories differ from the strictly stable ones. This work looks at three aspects of simple theories and uses them to give a better picture of the differences between the two classes. First, we look at the property of weakly eliminating hyperimaginaries and show that it is equivalent to forking and thorn- forking independence coinciding. Second, we look at the stable forking conjecture, a strong statement asserting that simple unstable theories have an essentially stable “core”, and prove that it holds between elements having SU-rank 2 and finite SU-rank. Third, we consider a property on indiscernible sequences that is known to hold in every stable theory, and show it holds on, at most, a subset of simple theories out of all possible first order theories. CONTENTS
ACKNOWLEDGMENTS ...... iii
CHAPTER 0: INTRODUCTION AND BACKGROUND ...... 1 0.1 Preliminaries ...... 5
CHAPTER 1: DWIP ...... 13 1.1 DWIP and the Weak Elimination of Hyperimaginaries ...... 13 1.2 WEHI and Thorn Forking ...... 15 1.2.1 Comparison with Ealy’s approach ...... 15 1.2.2 Non-forking independence has the intersection property . . 18 1.3 Equivalence of Forking and Thorn Forking Imply WEHI . . . . . 20
CHAPTER 2: SOME STABLE FORKING ...... 25 2.1 Symmetry of Stable Forking ...... 26 2.2 Quasidesigns have stable forking sometimes ...... 29 2.3 Stable Forking with Elements of SU-rank ≤ 2 ...... 33 2.4 A broader application of DWIP ...... 36
CHAPTER 3: SEQUENCE EXTENSIONS ...... 38 3.1 Common Extensions ...... 39 3.2 Graph Relation from No Common Extension ...... 44 3.3 Small-Tree Property ...... 47 3.4 Fans ...... 52 3.5 A Small Tree Example ...... 55
BIBLIOGRAPHY ...... 58
ii ACKNOWLEDGMENTS
I would like to thank all the people who helped me through my graduate years. Special thanks go to Cameron D. Hill and Sergei Starchenko. I would like to thank my advisor Steve Buechler for his assistance, and the Schmitt Foundation for a generous fellowship. Also, the loving support of Sarah, Chauncey, and my family.
iii CHAPTER 0
INTRODUCTION AND BACKGROUND
Simple theories are a class of elementary first-order theories which strictly in- clude the class of stable theories. They are characterized by the nice behavior of non-forking independence over arbitrary subsets. While their original uncovering was by Shelah (see [18]) in the course of looking for theories with nice combina- torial properties, interest waned until Kim (in [7]) showed that non-forking inde- pendence extended virtually unchanged from stable theories. Over the following years many properties of stable theories were reproved in the simple context. Two examples are the orthogonality of types and the supersimple/simple dichotomy. Other objects were extended, albeit with some changes. Types in simple theories have canonical bases, but the bases are represented as hyperimaginary elements instead of the imaginary elements used in the stable case. In some cases, such as supersimple or low simple theories, canonical bases can in fact be represented by imaginary elements. This behavior raises questions about how similar stable and simple theories are. There are obvious differences, such as the presence of unstable formulas in simple unstable theores, and the non-definability of types, since a type over a model in a simple theory can have unboundedly many global, non-forking extensions. Yet, because of the Type Amalgamation Theorem, extensions of that same type over a model to independent sets do have a common realization. Not
1 much is known about how a “generic” simple unstable theory differs from a stable theory. This work looks at three aspects of simple theories and uses them to give a better picture of the differences between the two classes. First, we introduce a property DWIP, and use it to show that a simple theory posesses it if and only if forking and thorn forking independence are equivalent and thorn forking has weak canonical bases. It is also shown that DWIP is equivalent to the weak elimination of hyperimaginaries. Second, we look at the stable forking conjecture, a strong statement asserting that simple unstable theories have an essentially stable “core”, and prove that it holds between elements having SU-rank 2 and finite SU-rank. Third, we consider a property on indiscernible sequences which is known to hold in every stable theory, and show it holds on, at most, a subset of simple theories out of all possible first order theories. The underlying concept among all these approaches is that of a canonical base of a type. In a stable theory, types have a unique smallest domain of definition, called the canonical base. The canonical base is not, in general, represented by elements in a structure, but rather by imaginary elements, That is, by names of classes of definable equivalence relations. In contrast, simple unstable theo- ries have canonical bases, but they cannot be represented as imaginary elements, instead hyperimaginary elements are needed. Much of the theory around hyper- imaginary elements was developed in order to provide a representation of canonical bases for forking-independence. The main issue with hyperimaginary elements is that they cannot be simply adjoined to a structure as imaginary elements can. Since a hyperimaginary element names a type-definable class, being able to name the class turns a closed set in the type space into a clopen set. Yet, some sub-
2 classes of simple theories have enough structure to allow canonical bases to be respresented by imaginary elements as in the stable case instead of hyperimagi- naries. This allows use to think of canonical bases as residing “in” the structure instead of beside it. One sufficient condition is for a theory has elimination of hyperimaginaries. The first comparison between stable and simple unstable theories comes from an early observation that although there are many examples of simple unstable theories, many of these seem to be essentially stable up to some kind of “noise.” So, in essence, there should not be much difference between the two classes, provided there is some way to define the word “noise,” and to account for it. A strong definition is to ask that every instance of forking in a simple unstable theory is always witnessed by a stable forking formula. The formalization of this is (see [10])
Definition 0.0.1. A simple theory T has stable forking if whenever q(x) is a complete type over a model M, A ⊂ M and q forks over A then there is is a stable formula ψ(x; b) ∈ q that forks over A.
The Stable Forking Conjecture then asks whether every simple theory has stable forking. The conjecture is unresolved—and not universally believed true— but it has been verified in several special classes including 1-based supersimple theories and stable theories expanded with a generic predicate. There have been many attempts at showing pieces of it (see [9], [10], [13], and [15]). Some involve streghthing the conjecture, others look outside elementary first-order theories to homogenous models. Peretz considers the context of ω-categorical supersimple theories, and there shows that there is always stable forking between elements of rank 2, and that between elements of rank 3, any counterexample to the conjecture
3 must exhibit a very specific type of indiscernible sequence. We extend Peretz’s results by dropping the condition of ω-categoricity, and then showing stable forking between an elements of SU-rank 2 and of finite SU-rank, so long as the theory has weak-elimination of hyperimaginaries. To draw a second distinction between stable and simple, we look at a notion of independence a priori weaker than forking independence called thorn forking independence. Onshuus introduced thorn forking (see [11]) and showed it to be a useful notion of independence in a broad class of theories, christened rosy theories, which include all stable, simple, and o-minimal theories. He also showed thorn forking is equivalent to Shelah forking in stable theories [12]. However, the rela- tionship between forking and thorn forking in simple theories remained an open question. In [4] Ealy showed the two notions of independence are equivalent in simple theories having elimination of hyperimaginaries. The chief obstacle is that in a simple theory canonical bases are represented by hyperimaginary elements. In contrast, thorn forking is related to algebraic closure, and is intimately tied to the imaginary elements of a theory. In fact, thorn independence is sensitive to whether one is working in T or T eq. Elimination of hyperimaginaries allows canon- ical bases to be represented as imaginary elements, and thus allows thorn forking to respect forking (and thus, to coincide). We observe that weak elimination of hyperimaginaries, while not quite preserving canonical bases, preserves enough for thorn forking to detect their presence. A theory having weak elimination of hyperimaginaries is equivalent to the theory having the property dependence witnessed by imaginaries (DWIP), which asserts that all instances of forking are witnessed by imaginary elements. This permits a “transfer of dependence” from hyperimaginaries to imaginaries to get the connection between forking and thorn
4 forking. Finally, some classes of theories can be characterized by properties on indis- cernible sequences. For example, well known work of S. Shelah shows a theory is stable if and only if every indiscernible sequence is in fact an indiscernible set [18]. L. Scow has characterized NIP theories as exactly the ones where every ordered graph indiscernible sequence is indiscernible, up to some technical conditions. In a stable theory a well known equivalence relation on indiscernible sequences is the property of having a common extension. A natural question is whether this condition is peculiar to stable theories, or whether it holds for other theories. In- discernible sequences in stable theories are related to canonical bases in the sense that for any stationary type p, the canonical base of p is inside the definable closure of any Morley sequence in p. This same property of canonical bases and Morley sequences also holds in simple theories, making it natural to to examine the indiscernible sequence equivalence relation in simple theories. The question is, what would cause two sequences which “should” have a common extension fail to have one. We formalize the question by defining a Y-formation and ask whether a Y-formation has a common extension.
0.1 Preliminaries
In what follows we assume a familiarity with common conventions of model theory. We assume all elements, and sets are inside in a large-enough, suitably saturated model M. When we write a formula ψ(¯x) we assume the finite tuplex ¯ contains all the free variables in ψ. Sometimes we will drop the bar and simply write x when we meanx ¯, a finite tuple of variables. We use the notation a ≡C b to mean a and b have the same type over C. Or, equivalently, that there is
5 an automorphism fixing C pointwise and taking a to b. Two references for this background material are [20] and [2]. To begin with we allow any ambient theory T . A partitioned formula ψ(¯x;y ¯) is unstable if there is an indiscernible sequence
(aibi : i < ω) such that |= ψ(ai; bj) if and only if i ≤ j. (Equivalently, <, >, or ≥). If a partitioned formula is not unstable, then it is stable. Note that the stability of a formula depends on the partitioning chosen. Stable formulas are closed under boolean operations, but not necessarily under quantification. A theory is stable if no formula consistent with the theory is unstable. If I is a linearly ordered index set and B is a set, then a sequence of, possibly infinite, tuples (ai : i ∈ I) is indiscernible over B if for every n, every increasing n- tuple from the sequence has the same type over B. That is, for indices i0 < ··· < in and j0 < ··· < jn,
ai0 ··· ain ≡B aj0 ··· ajn .
An indiscernible sequence is an indiscernible set if we allow the indices to have any ordering with respect to each other, so long as they are distinct. A fundamental result of Shelah [18] is
Fact 0.1.1. A theory T is stable if and only if every indiscernible sequence is an indiscernible set
Given a set B and a very long sequence of elements (ai : i < λ), we can extract a sequence (bi : i < ω) indiscernible over B such that for each (n + 1)-tuple
b0 ··· bn, there are indices i0 < ··· < in < ω with b0 ··· bn ≡B ai0 ··· ain using the Erd¨os-Rado theorem. If ψ(¯x;y ¯) is a partitioned formula and ¯b a parameter we say ψ(¯x; ¯b) k-divides over a set A if there is a set {bi : i < ω} with b0 = b and bi ≡A b such that
6 {ψ(¯x; bi): i < ω} is k-inconsistent. A formula ψ(¯x; ¯b) with parameter ¯b forks over A if ψ implies a disjunction of formulas (with possibly different parameters) each dividing over A. We define an independence relation from forking by saying a is independent from b over C if no formula in tp(a/Cb) forks over C, which we notate as a | b. ^ C Given a | b we ask whether b | a? This is called symmetry (of independence), ^ C ^ C and it need not hold because of the asymmetry between the left and right sides in the definition. We define the simple theories to be exactly the ones where ^| is symmetric. In addition to symmetry, non-forking independence satisfies the following prop- erties. Let A, B, C, D be small subsets of M.
• Right Transitivity: If D ⊂ C ⊂ B, A | B and A | C then A | B ^ C ^ D ^ D
• Local Character: For every B there is a subset B0 ⊂ B of cardinality |B0| < (|T | + |A|)+ such that A | B ^ B0
Defining simple theories by the symmetry of forking is a robust notion and there are many equivalent descriptions.
Fact 0.1.2 ([20] Theorem 2.4.7). The following are equivalent.
1. T is simple
2. Forking independence is symmetric
3. Forking independence is transitive
4. No formula has the tree property (to be defined below)
5. For every type p(¯x), there is a subset C of its domain of size |T | + |x¯| such that p does not fork over C. (i.e. forking independence has local character)
7 All of these properties relate back to dividing, but none do so as directly as the tree property, which can be thought of as “dividing gone bad.”
Definition 0.1.3. A partitioned formula ϕ(¯x;y ¯) has the k-tree property if there
¯ <ω is some tree of parameters (bη : η ∈ ω ), each having the same type, such that <ω ¯ for each initial segment η ∈ ω , the set {ϕ(¯x; bη_i): i < ω} is k-inconsistent, and for each path σ ∈ ωω, the set {ϕ(¯x; ¯b ): i < ω} is consistent. σi
If a formula ϕ(x; y) has the k-tree property then some finite conjunction V i ϕ(x; yi) will have the 2-tree property ([20] Proposition 2.8.8). For a type p, partitioned formula ϕ, and natural number k, the rank D(p, ϕ, k) measures the depth of the k-tree property which ϕ has on realizations of p. It is defined inductively by
• D(p, ϕ, k) ≥ 0 if p is consistent.
• D(p, ϕ, k) ≥ α + 1 if there is a parameter b such that ϕ(x; b) k-divides over the domain of p, and D(p ∪ {ϕ(x; b)}, ϕ, k) ≥ α.
• D(pϕ, k) ≥ λ for limit ordinal λ if D(p, ϕ, k) ≥ α for all α < λ.
Put D(p, ϕ, k) = ∞ if there is no ordinal bounding the rank from above. ϕ has the k-tree property iff for some type p we have D(p, ϕ, k) = ∞) We will use a variant of this rank in §3.3. Analogously to superstable theories, a theory is supersimple if no type forks over a finite subset of its domain. We define a forking rank under the model of U-rank in superstable theories. Let p be a type. We define SU(p) as an ordinal or ∞ by
• SU(p) ≥ 0 if p is consistent
8 • SU(p) ≥ α + 1 if there is some extension q ⊃ p such that q forks over the domain of p and SU(q) ≥ α.
• SU(p) ≥ λ for limit ordinals λ if SU(p) ≥ α for all α < λ.
Let SU(p) = ∞ if there is an infinite decreasing chain of forking extensions. If a and b are (possibly infinite) tuples, we write SU(a/b) for SU(tp(a/b)). A theory is supersimple if and only if every type has an ordinal SU rank. The Lascar inequalities for U rank extend unchanged to SU rank. For tuples a and b,
SU(b/a) + SU(a) ≤ SU(ab) ≤ SU(b/a) ⊕ SU(a)
The ⊕ denotes the commutative sum on ordinals. The key property of forking in a simple theory is the Type Amalgamation Theorem, which acts as a replacement to stationarity in stable theories. The base type is required to be an amalgamation base. Any type over a model or over a boundedly closed subset is an amalgamation base.
Fact 0.1.4. Let p(x) be an amalgamation base with domain A. If B and C are sets that are independent over A, and pB and pC are nonforking extensions of p to
B and C, respectively, then the partial type pB(x) ∪ pC (x) is consistent and does not fork over A.
The canonical base of a type p is the smallest set C such that realization of p are fixed by automorphism σ if and only if σ fixes C pointwise. In a breakthrough paper, Hart, Kim, and Pillay showed that for a type p(x, A) there is a type- definable equivalence relation Ep which is the transitive closure of the relation
“p(x, B) ∪ p(x, C) does not fork.” Then the hyperimaginary element AEp acts as the canonical base of p.
9 Indiscernible sequences take on a fundamental role in simple theories. A pair of tuples have the same Lascar strong type if they are equivalent under every bounded (having less than some fixed cardinal number of classes) equivalence relation. Kim showed that having the same Lascar strong type is type definable in a simple theory [8]. Equivalently, two tuples have the same Lascar strong type if and only if there is a sequence of automorphisms, each fixing a model, which take one to the other Stability theory makes much use of imaginary elements, each of which is a name for a class of a definable equivalence relation. Imaginary elements can be “added” to a theory by adding sorts to represent each definable equivalence relation, and this “eq” construction preserves all of the relevant properties of the original first-order theory. We can do the same thing for type-definable equivalence relations, calling the name of such an equivalence class a hyperimaginary. If E(¯x, y¯) is a type definable equivalence relation withx ¯ = (xi : i ∈ I) andy ¯ = (yi : i ∈ I) for some (possibly infinite) linearly ordered index set I, anda ¯ = (ai : i ∈ I) is some tuple of
eq elements from M thena ¯E is the hyperimaginary element representing the class of all tuples E-related toa ¯. It can be shown that any type-definable equivalence relation is equivalent to an infinite conjunction of equivalence relations defined by a countable type, so it suffices to add names only for the equivalence relations defined by countably-infinite conjunctions of formulas. A model M enriched with these names is denoted M heq. It’s not hard to see that M ⊂ M eq ⊂ M heq up to some natural and obvious identifications. Automor-
heq phisms of M lift to M , so if e =a ¯E is a hyperimaginary element we say an automorphism fixes e iff it fixes the E class ofa ¯ as a set. This is an annoyance of
10 dealing with hyperimaginaries: the formulas comprising the equivalence relation E do not need to be e-invariant. The lifting of automorphisms does let us extend definable closure and algebraic closure to M heq; we will use dcl and acl to refer to these extensions for this paper. There is also a new closure operator, the bounded closure operator written bdd(A), which contains exactly the elements that have only a bounded number of conjugates under automorphisms which fix A point- wise. It is easy to see for a set A of hyperimaginaries dcl(A) ⊂ acl(A) ⊂ bdd(A). By Compactness, if b ∈ bdd(A) is an imaginary element, then in fact b ∈ acl(A). Unfortunately, we cannot think of M heq as a first-order structure. The main issue is that type-definable sets are closed but not clopen, so the negation of such is not, in general, type-definable. However, a fragment of logic can be developed. Specifically, we can extend the notions of type and forking to hyperimaginaries. If a and b are hyperimaginaries then there are equivalence relations E(x, y) and ¯ ¯ F (w, z) and tuplesa ¯ = (ai : i ∈ I), b = (bi : i ∈ J) such that a =a ¯E, b = bF . The type tp(a/b) can then be expressed as the union of the partial types over ¯b,
∃wz[E(x, w) ∧ F (z, ¯b) ∧ ϕ(w0, z0)] for all ϕ such that the partial type is satisfied by a. In the equation above, w0 and z0 represent some finite subtuples of w and z. We could have chosen to base the type on any tuple ¯b∗ so long as ¯b∗ and ¯b are F -equivalent. This definition is the right one since it preserves the idea of types representing orbits of elements over a base set.
Fact 0.1.5. [6, Proposition 1.4] a0 realizes tp(a/b) if and only if there is an au- tomorphism f fixing b such that f(a) = a0.
11 From the notion of type one can derive the notion of an indiscernible sequence of hyperimaginaries. We then say a type p(x, b) = tp(a/b) divides over a set C if S there is a sequence (bi : i < ω) in tp(b/C) indiscernible over C such that i p(x, bi) is inconsistent. A type forks if it implies a finite disjunction of types that divide. Just as it is convenient to work in a theory which eliminates imaginaries, it is very nice to work with theories that eliminate hyperimaginaries. We say that a theory T eliminates hyperimaginaries (has EHI) if for every hyperimaginary e ∈ Mheq, there is a set of imaginaries ¯b ⊂ Meq such that dcl(¯b) = dcl(e). Finally, we say that a theory T has weak elimination of hyperimaginaries (has WEHI) if for every hyperimaginary e ∈ Mheq, there is a set of imaginaries ¯b ⊂ Meq such that bdd(¯b) = bdd(e).
12 CHAPTER 1
DWIP
This chapter introduces the property Dependence witnessed by imaginaries, DWIP, which is shown to be equivalent to having weak elimination of hyperimag- inaries. We then prove that for a theory, having DWIP is necessary and sufficient for forking and thorn forking independence to be equivalent, and for thorn forking to have weak canonical bases.
1.1 DWIP and the Weak Elimination of Hyperimaginaries
Definition 1.1.1. We say a simple theory has the Dependence Witnessed by Imaginaries Property (DWIP) if whenever a 6 | b for hyperimaginary elements a ^ C and b and set C then there is an imaginary element d ∈ acl(bC) such that a 6 | d. ^ C
Definition 1.1.2. We say a theory has weak elimination of hyperimaginaries
(WEHI) if for every hyperimaginary e ∈ Mheq, there is a set of imaginaries B ⊂ Meq such that bdd(B) = bdd(e).
If a given hyperimaginary e has a set of imaginaries B where bdd(B) = bdd(e), then we say e is weakly eliminated. This notion is parallel to that of elimination of hyperimaginaries.
13 Proposition 1.1.3. Suppose T is simple and has DWIP. Then for every hyper- imaginary e there is a set of imaginary elements D such that D ⊂ dcl(e) and e ∈ bdd(D). In particular, T has WEHI.
Proof. Fix an arbitrary hyperimaginary e. Let C = acl(e)∩Meq. For each element a ∈ C, let a0 be the imaginary element naming all finitely many conjugates of a over C. Let D be the set of all tuples a0 for a ∈ C. By construction D ⊂ dcl(e).
Claim 1.1.4. C = acleq(D).
Proof of claim. It is clear D ⊂ C so acleq(D) ⊂ acleq(C) = C. For the other direction, suppose c ∈ C. Then, by construction, there is a corresponding element c0 ∈ D, and there is an algebraic formula which says c is in the set named by c0. Hence c ∈ acleq(D).
Claim 1.1.5. A | e for any set A. ^ D
Proof of claim. Suppose not. Then there is some set A such that A 6 | e. By ^ D DWIP there is an imaginary element d ∈ acl(eD) ∩ eq such that A 6 | d. Be- M ^ D cause D ⊂ dcl(e), in fact acl(eD) = acl(e). Since d is imaginary d ∈ C, and so d ∈ acl(D). Hence A | d, contradicting our choice of d. ^ D
Now consider the case A = e. Then e | e, which implies e ∈ bdd(D). ^ D
Proposition 1.1.6. Weak elimination of hyperimaginaries implies DWIP.
Proof. Suppose T weakly eliminates hyperimaginaries, and a,b are hyperimaginar- ies such that a 6 | b. We need to find an imaginary element d ∈ acl(bC) such that ^ C a 6 | d. WEHI gives a set of imaginary elements B such that bdd(B) = bdd(b). ^ C Hence, a 6 | bdd(B), implying a 6 | B. Then by the finite character of forking, ^ C ^ C
14 there is a finite tuple d ∈ B such that a 6 | d. B is a set of imaginary elements, ^ C hence B ⊂ bdd(b) implies B ⊂ acl(b). Hence d ∈ B ⊆ acl(b) and we are done.
Corollary 1.1.7. A simple theory T has DWIP if any only if it has weak elimi- nation of hyperimaginaries.
Proof. Combine Propositions 1.1.3 and 1.1.6
1.2 WEHI and Thorn Forking
Ealy showed thorn-forking and forking coincide in simple theories with EHI by introducing a set of properties for an independence relation, and showing any independence relation satisfying these properties in a simple theory must be non- forking independence. Adler also shows the equivalence of forking and thorn- forking [1], albeit in a slightly different way. He introduces the intersection prop- erty and shows any non-trivial independence relation possessing it must be thorn forking. He then shows, under elimination of hyperimaginaries, that non-forking independence has the intersection property. We will follow Adler’s approach and show that under weak elimination of hyperimaginaries, non-forking independence has the intersection property.
In this section ^| will refer to non-forking independence (as opposed to an arbitrary independence relation).
1.2.1 Comparison with Ealy’s approach
Given elimination of hyperimaginaries, Ealy proved the equivalence of fork- ing independence and thorn-forking independence by introducing axioms for an independence relation and showing any relation satisfying them is non-forking in-
15 dependence. Then, thorn forking can be shown to also satisfy these axioms, and hence, be equivalent to forking. While both forking and thorn forking are defined on imaginary elements, the relation that follows is required to be on subsets of hyperimaginaries. The defini- tion of forking readily lifts to hyperimaginaries. To get thorn forking extended to hyperimaginaries, Ealy uses EHI. The elimination of hyperimaginaries allows one to replace hyperimaginary elements with sets of imaginary elements, and thorn forking extends by simply “moving” the hyperimaginaries to imaginary elements.
I heq Let ^| be an ternary relation on subsets of M satisfying the following prop- erties:
1. Anti-reflexivity: a 6∈ bdd(C) =⇒ a 6 |I a for all a, C ^ c
2. Extension: For any A, B, C there is some A0 ≡ A such that A0 |I B C ^ C
3. Symmetry: A |I B =⇒ B |I A for all sets A, B, C ^ C ^ C
4. Transitivity: A |I BD ⇐⇒ A |I B ∧ A |I D, for all sets A, B, C, D ^ C ^ C ^ CB
+ 5. Local Character: For any A, B there is B0 ⊂ B with |B0| < |T | such that A |I B ^ B0
6. Finite Character: For every A, B, C, sets A |I B ⇐⇒ A |I B for all ^ C 0 ^ C
finite subsets A0 ⊂ A
I If we only required ^| to be defined on imaginaries, then thorn forking indepen-
I dence would satisfy all the properties. Having ^| defined on hyperimaginaries is enough to show it must be the relation of non-forking independence.
I Theorem 1.2.1 ([4], Theorem 2.3.3). If T is simple then ^| and ^| are equiva- lent.
16 I The proof begins by first showing that ^| dependence always implies ^| de- pendence. Then, supposing the two relations are different, an element of infinite D(−, φ, k) rank is exhibited, for some formula φ and natural number k. A key step is a partial type amalgamation result on I-independent extensions over a base set C.
Lemma 1.2.2 ([4], Corollary 2.3.2). If p(x; b, C) does not I-fork over C then there are b1, b2 |= tp(b/C) such that p(x, b1,C) ∪ p(x, b2,C) is consistent and has
I a realization a such that a | b2. ^ b1C
I The proof of Theorem 1.2.1 then begins by assuming ^| and ^| are not equiv- alent, so there are witnessing elements a, b, c with a 6 | b and a |I b. Let d = ^ c ^ c Cb(a/bc). A short argument shows d∈ / bdd(ab), implying there are I-independent realizations d1, d2 of tp(d/ac) over ac. Then a 6 | d2, yet Lemma 1.2.2 gives ^ cd1
I a | d2. Induction then gives an arbitrarily long chain of types such that each ^ cd1 is a forking extension of the previous one. This gives a contradiction to simplicity,
I and hence ^| and ^| are equivalent. A theory having elimination of hyperimaginaries implies thorn forking inde-
I pendence satisfies the properties for ^| , and hence, is equivalent to non-forking in- dependence. In fact, weak elimination of hyperimaginaries is sufficient to give this extension, from the equivalence between DWIP and WEHI. Once this is noted, Ealy’s proof works without any modifications under the weaker assumption of WEHI.
Corollary 1.2.3. If T is simple has WEHI, then forking and thorn-forking inde- pendence are equivalent.
17 1.2.2 Non-forking independence has the intersection property
We will now present a more hands-on proof to show non-forking independence and thorn-forking independence are equivalent, using Adler’s approach [1]. Adler works in the general context of abstract ternary relations on subsets of a model, similar to Ealy.
* * Definition 1.2.4. Let ^| be a ternary relation on subsets of a model. ^| is an in- dependence relation if the following properties hold. For the following, A, B, C, D are arbitrary sets
• Invariance: A |* B and A0B0C0 ≡ ABC =⇒ A0 |* B0 ^ C ^ C0
• Monotonicity: A |* B, A0 ⊂ A, B0 ⊂ B =⇒ A0 |* B0 ^ C ^ C
• Base Monotonicity: If D ⊂ C ⊂ B and A |* B then A |* B ^ D ^ C
• Transitivity: If D ⊂ C ⊂ B, B |* A and C |* A then B |* A ^ C ^ D ^ D
• Normality: A |* B =⇒ AC |* B ^ C ^ C
• Extension: If A |* B and Bˆ ⊃ B then there is A0 ≡ A such that ^ C BC A0 |* Bˆ ^ C
• Finite Character: If A |* B for all finite A ⊂ A then A |* B 0 ^ C 0 ^ C
• Local Character: For every A there is cardinal κ(A) such that for any B,
there is a subset C ⊂ B of cardinality |C| < κ(A) such that A |* B ^ C
• Anti-Reflexivity1: a |* a implies a ∈ acl(B) ^ B 1Adler does not include this as a main axiom. However, he only works with what he calls strict independence relations, which are those having anti-reflexivity.
18 Adler examines a variant of canonical bases in this general context. A nec- essary condition for an independence relation to have “canonical bases” is that independence needs to be closed under intersection of such bases.
* Definition 1.2.5. We say an independence relation ^| has the intersection prop-
* * erty if whenever C1,C2 ⊆ B such that A | B and A | B then ^ C1 ^ C2
* A ^| B. eq eq acl (C1)∩acl (C2)
Adler makes a distinction between a relation having the intersection prop- erty and having both the intersection property and anti-reflexivity, which he calls canonical. Since we requires all independence relations to be anti-reflexivity, the distinction is unimportant here. An independence relation having the intersection property implies the inde- pendence relation is in fact thorn forking.
* Theorem 1.2.6 ([1], Theorem 3.3). If ^| is a non-trivial independence relation
* with the intersection property, then ^| is thorn forking.
The converse of the theorem does not hold: there are (non-simple) theories where thorn forking does not have the intersection property [1]. The power of having the intersection property is that it requires the indepen- dence over the intersection of algebraic closures. Thus, it is not clear whether forking-independence even has it since the canonical bases for non-forking inde- pendence are only represented by hyperimaginary elements. However, since DWIP always enforces that forking witnessed by imaginary elements, canonical bases and imaginary elements are intertwined.
19 Theorem 1.2.7. Suppose T is simple and has DWIP. Then forking has the in- tersection property.
Proof. Choose any sets A and B. Suppose there are subsets C1,C2 ⊆ B with A | B and A | B. We need to show A | B. ^ C1 ^ C2 ^ acl(C1)∩acl(C2) Since T is simple there is hyperimaginary e = Cb(A/B) and e ∈ bdd(B).
Moreover, e ∈ bdd(C1) ∩ bdd(C2), by using properties of canonical bases. Using
DWIP we get a set of imaginary elementse ˆ ⊂ dcl(e) ∩ Meq such that e ⊂ bdd(ˆe) and for all sets U, U | e. In particular, A | e. Sincee ˆ ⊂ dcl(e) ⊂ bdd(C ) ^ eˆ ^ eˆ 1 ande ˆ is a tuple of real elements,e ˆ ⊆ acl(C1). The same reasoning also shows eˆ ⊆ acl(C2). Thuse ˆ ⊆ acl(C1) ∩ acl(C2). Moreover, transitivity of forking gives
A ^| e ∧ A ^| B =⇒ A ^| eB. eˆ e eˆ
And so, A | B sincee ˆ ∈ acl(C1) ∩ acl(C2). ^ acl(C1)∩acl(C2)
Corollary 1.2.8. If T is simple and has DWIP, forking is equivalent to thorn forking and thorn forking has the intersection property.
Proof. Combine Theorems 1.2.7 and 1.2.6. Since forking has the intersection property and is equivalent to thorn forking, thorn forking also has the intersection property.
1.3 Equivalence of Forking and Thorn Forking Imply WEHI
The results in this section address the weak elimination of hyperimaginaries, specifically we show the converse of Corollary 1.2.8.
20 Theorem 1.3.1. Let T be simple. If forking and thorn-forking are equivalent (on imaginaries) and thorn forking is canonical, then T has admits weak elimination of hyperimaginaries.
The main issue is that we only assume the equivalence of the two notions on imaginary elements. We will work in both M eq and M heq, where M is a sufficiently large, sufficiently saturated monster model of T . We will use acleq and dcleq to indicate when the closure operations are on M eq, and assume the unadorned versions acl and dcl always include hyperimaginaries. The theorem is done as a sequence of lemmas. The first is easy, and is essen- tially identical to Lemma 3.6.4 in Wagner.
Lemma 1.3.2. If T weakly eliminates every hyperimaginary arising as a canonical base, then T weakly eliminates every hyperimaginary.
Proof. Consider an arbitrary hyperimaginary aE (where a may be an infinite tu- ple here) and E is a type-definable equivalence relation. Let d = Cb(a/aE). Then, a | a . By hypothesis, there is a set of imaginary elements C such ^ d E that bdd(C) = bdd(d), so also a | a . Since a ∈ dcl(a), we have a ∈ ^ C E E E bdd(C), but also, by definition d ∈ dcl(aE), so C ⊂ bdd(d) ⊂ bdd(aE). Hence
bdd(C) = bdd(aE), and aE is weakly eliminated.
A possible objection to the above proof is that when we talk of a canonical base of a type, we are thinking of a type over a domain of imaginary elements and not a hyperimaginary element as used in the previous proof. This is not an issue because of the following lemma.
Lemma 1.3.3. If T weakly eliminates canonical bases of types over imaginaries, then T also weakly eliminates canonical bases of types over hyperimaginaries.
21 Proof. Let p(x, e) be a type over hyperimaginary e. Then p is equivalent to a par- tial type π(x, eˆ) over a possibly infinite imaginary tuplee ˆ that is a member of the equivalence class named by e. Let p0 ∈ S(ˆe) be a completion of π that does not fork over e. Then p0 is a non-forking extension of p, and so bdd(Cb(p0)) = bdd(Cb(p)). Since p0 is defined over a set of imaginary elements, the hyperimaginary d = Cb(p0) is weakly eliminated. Hence the canonical base of p, being interbounded with d, is also weakly eliminated.
In the presence of the intersection property, there is a smallest algebraically closed set over which a type does not fork. We call this set the weak canonical base.
Definition 1.3.4. Let p be a type over B ⊂ M eq. The weak canonical base, WCb(p) is the smallest algebraically closed subset (of imaginaries) C ⊂ B such that p does not fork over C.
The following properties for weak canonical bases parallel those for canonical bases.
Proposition 1.3.5 ([1], Exercise 3.21). For a tuple a and sets B,C the following are equivalent.
1. a¯ | B ^ C
2. WCb(¯a/BC) ⊂ acl(C)
3. WCb(¯a/BC) = WCb(¯a/C)
Adler ties the existence of weak canonical bases to forking having the intersec- tion property.
22 Theorem 1.3.6 ([1], Theorem 3.20). ^| has the intersection property iff every type over an algebraically closed subset has a weak canonical base.
Our goal is to show weak canonical bases, when they exist, behave enough like canonical bases so as to eliminate them.
Proposition 1.3.7 ([20], Theorem 2.5.4). Let π(x, a) be a partial type over Aa that does not fork over A. If (ai : i < ω) is a sequence in tp(a/A) indiscernible S over A, then i π(x, ai) is consistent and does not fork over A.
Lemma 1.3.8. If ^| has the intersection property, then for each type p over imaginary elements, the hyperimaginary e = Cb(p) is weakly eliminated.
Proof. Let p(x) be a type over a set B of imaginaries. Let D = WCb(p), so D ⊂ B is a set of imaginary elements and p does not fork over D. On the other hand, there is a canonical base e = Cb(p) which exists as a hyperimaginary element. By definition, since p does not fork over D, we have e ∈ bdd(D). We wish to reverse this and show D ⊂ bdd(e). Aiming for a contradiction, suppose D 6⊂ bdd(e). Then there is an e-indiscernible sequence (Di : i < ω) of pairwise distinct realizations of tp(D/e) with D0 = D. Let p0(x, D) be the restriction of p to D. Since p does not fork over e, neither
0 0 0 0 does p (x, D). For i < ω, p (x, Di) is a conjugate over e of p (x, D). Since p (x, D)
0 does not fork over e, the union q(x) = ∪p (x, Di) is consistent and for each i, q is
0 a non-forking extension of p (x, Di). In particular this means C = WCb(q), the weak canonical base of q, is contained in Di for each i. In fact C = Di for each
0 i since WCb(p (x, Di)) = Di. And so Di = C = Dj for every pair i, j < ω. But this contradicts the sequence consisting of distinct realizations of tp(D/e). Hence D ⊂ bdd(e), and e is weakly eliminated.
23 Finally, we come to the proof of Theorem 1.3.1, stated at the beginning of the section.
Proof of Theorem 1.3.1. Assume the hypothesis. Then forking has the intersec- tion property, and by Lemma 1.3.8, canonical bases of types over imaginary ele- ments are weakly eliminated. Then by Lemmas 1.3.3 and 1.3.2, all hyperimagi- naries are weakly eliminated.
Corollary 1.3.9. Let T be simple. Then T has weak elimination of hyperimag- inaries if and only if forking is equivalent to thorn forking, and thorn forking is canonical.
Proof. Combine Theorem 1.3.1 and Corollary 1.2.8.
The result below summarizes the results in this chapter.
Theorem 1.3.10. Let T be a simple theory. The following are equivalent.
1. T has WEHI
2. T has DWIP
th th 3. ^| = ^| and ^| is canonical in T .
24 CHAPTER 2
SOME STABLE FORKING
This chapter looks at stable forking and finds it between elements having low SU-rank. The motivation was Peretz’s finding of stable forking between elements of SU-rank 2 in ω-categorical theories [13]. We manage to remove the assumption of ω-categoricity as well as allowing one of the elements to have finite SU-rank, instead of just rank 2. But, this has the cost of assuming the theory has weak elimination of hyperimaginaries, and only working over acl(∅) as the base set. The inductive argument centers around reducing the rank of the finite ranked element until it has SU-rank 1, at which point a stable forking formula is easy to find. The problem is finding elements of smaller SU-rank to do this reduction, since the smaller ranked elements must lie in the algebraic closure of the initial element. In fact, it may not always be possible. To this end, we identify the situations where the reduction cannot happen (where z holds), and then show that in every case we get a quasidesign over ∅. We then show quasidesigns over ∅ themselves always have stable forking. In [17] Shami showed that stable forking is symmetric under the additional assumption that strong types and Lascar strong types coincide. While supersim- plicity implies the equivalence of Lascar strong type and strong type, we are only working under the assumption of WEHI. In private correspondence, Shami has
25 shown that this assumption is unnecessary. We present Shami’s proof of the more general result in §2.1.
2.1 Symmetry of Stable Forking
This section presents a proof that stable forking is symmetric in all simple theories. It builds on a foundation laid in [17] where stable forking symmetry is proved for theories where Lascar strong types are equivalent to strong types. We will recall the essentials of the argument in Shami’s paper, but only present a proof of the new result. All of the results in this section are due to Shami; the arguments have been filled in around sketches provided by Shami. Here is a rough outline of the original proof. First, one establishes that for stable formulas, generic satisfiability is unique for parameters which have the same Lascar strong type.
Fact 2.1.1. [17, Claim 6.5] Let T be simple, and let φ(x, y) ∈ L be stable. Assume a | b and a0 | b and that Lstp(a/A) = Lstp(a0/A). Then |= φ(a, b) if and only ^ A ^ A if |= φ(a0, b).
One uses this fact to identify an alternate definition of forking (more akin to forking in a stable theory) with the usual one for simple theories.
Definition 2.1.2. Let φ(x, y) be a formula, and let A ⊆ B ⊂ Meq. Then
p ∈ Sφ(B) does not fork in the sense of LS if, for some model M containing
0 B, some p ∈ Sφ(M) extending p is definable over acl(A).
Fact 2.1.3. [17, Lemma 6.6] Assume T is a simple theory with Lstp = stp, and
eq let φ(x, y) ∈ L be a stable formula. Let a ∈ M and A ⊆ B ⊂ M. Then tpφ(a/B) does not fork over A in the sense of LS if and only if tpφ(a/B) does not fork over A.
26 Provided Lstp = stp, it is a short step from this to the symmetry of “stable non-forking independence,” from which the symmetry of stable forking is fairly obvious. Throughout Shami’s argument, the assumption that Lstp = stp appears only so that Fact 2.1.1 may be used. Thus, to extend his argument to arbitrary simple theories, it suffices to prove an analog of Fact 2.1.1 for strong types in place of Lascar strong types. Strangely, the proof of this extension actually uses its precursor.
Lemma 2.1.4. Let E(x, y) be a bounded, co-type definable equivalence relation. Then E is a definable, finite equivalence relation.
Proof. Suppose E is as in the hypothesis. Let r(x, y) be a partial type defining ¬E. If E had an infinite number of equivalence classes, then we could build an indiscernible sequence where each 2-type realizes r. The sequence would then imply the number of E classes is unbounded, contradicting the hypothesis. Thus E has only a finite number of classes. Let a1, . . . , an for some n < ω consist of a single V representative from each class. Then the type i≤n r(x, ai) is inconsistent, so there V is a formula ψ(x, y) ∈ r such that i≤n ψ(x, ai) is inconsistent, by Compactness. V W Observe that ¬∃x( i≤n ψ(x, ai)) is equivalent to ∀x( i≤n ¬ψ(x, ai)), so for every x there is some j such that ¬ψ(x, aj). Since ψ ∈ r, ¬ψ(x, y) |= E(x, y). Let
^ θ(x, y) ≡ (¬ψ(x, ai) ↔ ¬ψ(y, ai)) . i≤n
We will show θ defines E. First suppose θ(b, c). For some j we have ¬ψ(b, aj), and θ entails ¬ψ(c, aj). Thus E(b, aj) and E(c, aj) so by transitivity E(b, c). Conversely, suppose E(b, c). If ¬θ(b, c) then there is some index i such that
(without loss of generality) ¬ψ(b, ai) ∧ ψ(c, ai) holds. There is another index j
27 such that ¬ψ(c, aj). Then we have E(b, ai), E(b, c) and E(c, aj). By transitivity
E(ai, aj). But we chose ai and aj to be in different classes. The contradiction proves the claim and the lemma.
Lemma 2.1.5. Let T be simple. Let φ(x, y) ∈ L be stable. Assume a | b and ^ A a0 | b and that stp(a/A) = stp(a0/A). Then |= φ(a, b) if and only if |= φ(a0, b). ^ A
Proof. Given a complete type q(x) define the equivalence relation Eq by
φ 0 0 0 Eq (a, a ) ⇐⇒ for every b |= q with b ^| aa , |= φ(a, b) ↔ φ(a , b).
φ The complement of Eq is defined by a partial type over A. By Fact 2.1.1, if
0 φ 0 φ Lstp(a) = Lstp(a ), then Eq (a, a ) holds; this implies Eq is refined by the equality of Lascar strong type, so it is bounded. A bounded, co-type-definable equivalence
φ relation is a finite, definable equivalence relation by Lemma 2.1.4, so Eq is an A-definable finite equivalence relation. Now, suppose stp(a/A) = stp(a0/A), b | a and b | a0. Put q = tp(b/A). ^ A ^ A eq eq 0 φ By definition, tp(a/ acl (A)) = tp(a/ acl (A)), so a, a must be in the same Eq - class. Let a00 realize stp(a/A) such that a00 | aa0b. By transitivity, aa00 | b ^ A ^ A and a0a00 | b. Then, since Eφ(a, a00) and Eφ(a0, a00) we have φ(a, b) ↔ φ(a00, b) ↔ ^ A q q φ(a0, b), as desired.
The remainder of Shami’s argument (Lemma 6.6 to the end of section 6 of [17]) goes through essentially unchanged except for substituting our Lemma 2.1.5 for Fact 2.1.1.
Theorem 2.1.6. Let T be simple. If a | b and there is a stable formula ψ(x; y) ^ C such that ψ(x; b) forks over C, then there is a stable formula ϕ(x; y) such that ϕ(x; a) forks over C.
28 2.2 Quasidesigns have stable forking sometimes
The eventual argument will require there to be an indiscernible sequence in tp(b/aC) such that a is not in the algebraic closure of the sequence with C. Such sequences may not always exist. To this end, we will find some sufficient conditions for when they do. The property we are concerned about is
(z) a∈ / acl(bC), b∈ / acl(aC), and for every non-constant aC-indiscernible sequence I in tp(b/aC), a ∈ acl(IC).
We will speak of elements (a, b, C) satisfying property z in the natural way, and we will denote this condition by z(a, b, C).
Proposition 2.2.1.
1. In the definition of z it is equivalent to require for every infinite set D of distinct realizations of tp(b/aC) we have a ∈ acl(DC).
2. The property z is co-type-definable. That is, there is a partial type Ω(x, y, z) such that for all a, b, C (with |a| = |x|, |b| = |y|, |C| = |z|) we have |= Ω(a, b, C)
if and only if (a, b, C) do not satisfy z.
3. If z(a, b, C) holds then there are formulas δ(x, z), ζ(y; xz), θ(x; zy0 . . . yn) such that for some finite c ⊂ C
(a) δ(x, z) ∈ tp(ac)
(b) ζ(y; ac) ∈ tp(b/ac)
(c) θ(x; zy¯) is algebraic in x
(d) The following entailment holds
" ! # ^ ^ δ(x, z) ` ∀y0 . . . yn yi 6= yj ∧ ζ(yi; xz) → θ(x; zy0 . . . yn) i 29 Proof. (1) If a is algebraic in every infinite set of realizations of tp(b/aC) and C, then certainly a is algebraic over any indiscernible sequence in tp(b/aC) and C. Conversely, if there is some infinite set B of realizations of tp(b/aC) for which a is not algebraic over BC, then we can, using Compactness, get an arbitrarily 0 0 large set B = (bα : α < λ) of realizations of tp(b/aC) such that a∈ / acl(B C). From B0 we may extract a sequence I which is indiscernible over aC. But then a∈ / acl(IC). (2) Since the statement x∈ / acl(z) is a partial type in x and z, the desired type says that there are infinitely many elements realizing tp(y/xz) over which x is not algebraic. " # ^ ^ ∃y0y1 ... x∈ / acl(zy0y1 ... ) ∧ tp(yi/xz) = tp(y/xz) ∧ yi 6= yj i i6=j (3) If z(a, b, C) holds then the partial type in (2) is inconsistent. Hence there is some finite c ⊂ C, a formula ζ(y, ac) ∈ tp(b/aC), and an algebraic formula θ(x; zy0y1 . . . yn), for some n < ω, such that " ! # ^ ^ tp(ac) ` ∀y0 . . . yn yi 6= yj ∧ ζ(yi; ac) → θ(a; cy0 . . . yn) i The right hand side is a formula, hence there is a formula δ(x, z) ∈ tp(ac) which implies it, by Compactness. The formula in part (3)d allows us to recover algebraicity from some finite number of elements which individually satisfy only a finite subset of tp(b/ac). This will be important for what follows. In fact, a direct application of part (3) above shows that over the empty set there is always stable forking. 30 Proposition 2.2.2. If z(a, b, ∅), holds then there is a stable formula ψ(x; b) ∈ tp(a/b) which forks over ∅. Proof. Suppose z(a, b, ∅) holds. From Proposition 2.2.1, there are formulas δ(x), ζ(y, x), and θ(x; y1 . . . yn) for some n < ω. Since z holds over the empty set, the “finite tuple” in question is null; hence, the formulas reduce to the following entailment. " ! # ^ ^ δ(x) ` ∀y0 . . . yn yi 6= yj ∧ ζ(yi; a) → θ(a; y0 . . . yn) (2.1) i Consider the formula ψ(x; b) ≡ δ(x)∧ζ(b; x) ∈ tp(a/b). We will show ψ is a stable formula and ψ(x; b) forks over ∅. Stable: Suppose ψ is not stable, and let I = (diei)i∈Z be an indiscernible sequence witnessing this by |= ψ(di; ek) iff i ≤ k. (Note that this sequence is nec- essarily non-constant.) Then δ(di) holds for all i. Consider the element d−1. Since ψ(d−1; ek) holds for 1 ≤ k ≤ n, so does ζ(ek; d−1). Hence, from the entailment in equation (2.1), θ(d−1; e1e2 . . . en) is true, which witnesses d−1 ∈ acl(e1e2 . . . en). This contradicts that I is an indiscernible sequence. Hence, ψ is stable. Forking: Suppose ψ(x, b) does not fork over ∅. Let r(x, b) be a non-forking extension of the partial type {ψ(x, b)}, and let c |= r(x, b) realize it, so c ^| b. Let I = (bi : i < κ) be an indiscernible sequence such that b0 = b and 0 κ is sufficiently large for the remainder. Note that r (x, I) = {r(x, bi): i < κ} 0 0 0 0 is consistent, and we may choose c |= r (x, I) such that c ≡b c and c ^| I. As, by assumption, κ is large enough, there is an infinite subset I0 ⊂ I that is 0 indiscernible over c . Without loss of generality, we assume that I0 = (bi : i < ω). 0 0 Now, note that ψ(c , bi) for each i < ω, so in particular, |= ζ(c , b0, ..., bN ). It 0 0 0 follows that c ∈ acl(b0...bN ), and since c ^| b0...bN , we have c ∈ acl(∅). This 31 implies c ∈ acl(∅). Now, if every non-forking extension tp(c∗/b) of {ψ(x, b)} induces c∗ ∈ acl(∅), then |= ψ(x, b) =⇒ x ∈ acl(b). Since ψ(x, b) ∈ tp(ab), this contradicts the assumption that a∈ / acl(b). Thus, ψ(x, b) forks over ∅. A configuration related to z is the quasidesign. For a set C, a quasidesign over C is a partial type r(x, y) over C such that for any a, b |= r we have a∈ / acl(bC), b∈ / acl(aC), and for all distinct b1, b2 the set 0 0 0 {a : |= r(a , b1) and |= r(a , b2)} is finite. A pseudoplane is a quasidesign where also for all distinct a1, a2 the 0 0 0 set {b : |= r(a1, b ) and |= r(a2, b )} is also finite. A theory omits quasidesigns (pseudoplanes) if there is no partial type that is a quasidesign (pseudoplane). Omitting quasidesigns is equivalent to 1-basedness. Fact 2.2.3 (see [14]). The following are equivalent. 1. T is 1-based 2. T omits quasidesigns (over ∅) 3. T omits pseudoplanes (over ∅) The salient property of quasidesigns which relates them to z is the following Fact 2.2.4 (see [14]). Suppose r(x, y) is a quasidesign over C, b is an element and (ai : i < ω) is a set of elements such that r(ai, b) holds for all i. Then b ∈ acl(C(ai)i<ω). Proposition 2.2.5. If r(x, y) is a quasidesign over C, and a, b are elements such that r(a, b) holds, then z(b, a, C) holds. 32 Proof. Let r be a quasidesign and |= r(a, b). By definition a∈ / acl(Cb) and b∈ / acl(Ca). To exhibit z(b, a, C) we only need to show that for any indis- cernible sequence I in tp(a/Cb) we have b ∈ acl(CI). Yet, for any such sequence I = (ai : i < ω), we have |= r(ai, b) for all i. Hence b ∈ acl(CI). 2.3 Stable Forking with Elements of SU-rank ≤ 2 We now show that forking between a finite-rank element and one of SU-rank ≤ 2 over ∅ is always witnessed by a stable formula. The main theorem of this section relies on DWIP results from Chapter 1. Since it treats elements of SU- rank exactly 2, we begin by handling ranks < 2. Since the first results allow an arbitrary base set, C, we include it. Proposition 2.3.1. Let C ⊂ M, and let a, b ∈ M be finite tuples. If SU(b/C) = 1 and a 6 | b then there are stable forking formulas in both tp(a/bC) and tp(b/aC). ^ C Proof. From a 6 | b, we have SU(b/aC) < SU(b/C) = 1, so b ∈ acl(aC) \ acl(C). ^ C Let θ(y; ac) ∈ tp(b/aC) be an algebraic formula. Then θ(y; ac) forks over C because b∈ / acl(C), and it is stable because it is algebraic. The stable forking formula inside tp(b/aC) then follows from Theorem 2.1.6. We next show stable forking is passed “upward” through algebraic closure. It is similar to the result of Kim [9] that if E(y; z) is a finite definable equivalence relation and ϕ(x; y) is any formula then ∃y[ϕ(x; y) ∧ E(y; z)] is stable. For this one claim we do not need to assume T is simple. Lemma 2.3.2. Let T be an arbitrary theory. Suppose ζ(x; y) is a stable formula and θ(y; zw) is algebraic in y. Let ψ(x; zw) be the formula ∃y[ζ(x; y) ∧ θ(y; zw)]. Then, 33 1. ψ(x; zw) is stable; 2. if there are elements a, b, c, d and a set D containing d such that a |= ζ(x; c), ζ(x; c) forks over D, and θ(x; bd) isolates tp(c/bD) (so c ∈ acl(bD)) then ψ(x; b) forks over D. Proof of (1). Towards a contradiction, suppose ψ(x; zw) is unstable. Let (aibici : i < ω + ω) be an indiscernible sequence witnessing the order property— i.e. |= ψ(ai; bjcj) iff i ≤ j. For i < ω, let di be the element witnessed by the existential in ψ(ai; bωcω). Since θ(y; bωcω) is algebraic, there are only finitely many 0 possible di, so by the pigeonhole principle at least one, d , is repeated infinitely of- 0 ten. As ζ is stable and (ai)i<ω+ω is indiscernible, the set {i < ω+ω : |= ζ(ai; d )} is either finite or cofinite, and by our choice of d0, it must be cofinite. Consequently, 0 there are indices k > ω such that |= ζ(ak; d ), and this entails |= ψ(ak; bωcω), a contradiction . Proof of (2). Let (ci)i We restate the previous lemma to have a more useful form. This corollary is what we mean by stable forking passing “upward” in algebraic closure. The stable forking of a with d is passed to a and b. Corollary 2.3.3. Let T be an arbitrary theory. Suppose a and b are tuples where tp(a/Cb) forks over C. Moreover, suppose there is an element d ∈ acl(Cb) such that tp(a/Cd) forks over C via a stable formula. Then tp(a/Cb) also contains a stable forking formula. 34 Theorem 2.3.4. Let T be a simple theory with DWIP. Let a, b ∈ M, and assume that SU(a) = 2 and SU(b) < ω. Then, if a ^6 | b, there is a stable formula in tp(a/b) which forks over ∅. Proof. Suppose a ^6 | b. The proof is by induction on the SU rank of b. If SU(b) = 1 then Proposition 2.3.1 produces a stable forking formula in tp(a/b). Now suppose SU(b) = r < ω and the proposition holds for all elements having smaller rank. Let ϕ(x; b) be an forking formula in tp(a/b). Either z(a, b, ∅) holds or it doesn’t. If it does, then there is a stable forking formula inside tp(a/b). If it doesn’t, then there is a sequence I = (bi : i < ω) con- taining b which is indiscernible over a and for which a 6∈ acl(I). The dependence a ^6 | b implies a ^6 | I which gives the rank inequalities 1 ≤ SU(a/I) ≤ SU(a/b) < SU(a) = 2. Hence, SU(a/I) = SU(a/b), and so a | I. Put e = Cb(a/I). Since ^ b a and I are independent, e ∈ bdd(b). In fact, we could have done the above rank argument with any b0 ∈ I to give e ∈ bdd(b0). However, b∈ / acl(e), since b ∈ acl(e) would imply b ∈ acl(b0) for any b0 ∈ I, which contradicts both b and b0 being in the same indiscernible sequence I. Hence SU(b/e) > 0 and SU(e/b) = 0. The Lascar inequalities then show SU(e) < SU(b): SU(e) < SU(b/e) + SU(e) ≤ SU(eb) ≤ SU(e/b) ⊕ SU(b) = SU(b) Both a 6 | I and a | I imply a 6 | e by transitivity. DWIP gives a finite, real ^ ^ e ^ tuple d ∈ acl(e) with a ^6 | d. The induction hypothesis then provides a stable forking formula inside tp(a/d) since SU(d) ≤ SU(e) < SU(b). Then because d ∈ acl(b) Corollary 2.3.3 gives a stable forking formula in tp(a/b). 35 The main obstacle to extending the proof to elements a with SU-rank larger than 2 is that it is no longer possible to force Cb(a/I) to lie inside the bounded closure of a single element of I. Indeed, in general the canonical base would be inside the bounded closure of I. 2.4 A broader application of DWIP The main observation in Theorem 2.3.4 is that when z does not hold, a and b are in 1-based position with each other by virtue of a having SU-rank 2. This suggests that the same proof should work in a 1-based theory, and it does, as shown in this section In some ways, stable forking in simple 1-based theories was already known since Kim showed it to hold in simple theories having elimination of hyperimaginaries [9]. This proof is more limited, since we have the same rank requirements on a and b as before. However, we only require the weak elimination of hyperimaginaries. We first need a lemma showing we can get an indiscernible sequence as we required. Lemma 2.4.1. Suppose T is a simple, 1-based theory. If a,b are elements such that a 6 | b for some set C then there is sequence I = (b : i < ω) indiscernible ^ C i over aC such that b0 = b and satisfying a | I for all i < ω. ^ biC Proof. Let a,b,C be as in the statement of the lemma. We will consider “negative” + + ordinals for indexing purposes, and let J = (bi : −|T | ≤ i ≤ |T | ) be a Morley sequence in tp(b/aC). By using an automorphism, we may assume b0 = b. Let + + L = (bi : 1 ≤ i ≤ |T | ) and K = (bi : −|T | ≤ i ≤ −1). For the moment we will focus on L. By 1-basedness, L is Morley over b0C. For some subset D ⊂ L such ∗ ∗ that |D| ≤ |T |, we have a | L. Let L = L \ D. Since L is Morley over b0C, ^ b0CD D | L∗. Apply transitivity to get a | L∗. In the same way we find a subset ^ b0C ^ b0C 36 ∗ ∗ ∗ ∗ K ⊂ K such that a | K . There is a subset D ⊂ K such that a | ∗ K , ^ b0C ^ b0CL D ∗ ∗ ∗ ∗ by replacing K with K \ D we may assume a | K b0L (note that in either ^ b0C ∗ + 0 ∗_ _ ∗ 0 case we have |K | = |T | ). Let I = K b0 L . The sequence I is indiscernible 0 0 over aC, which implies for each bi ∈ I, a | I . Taking I = (bi : 0 ≤ i < ω) ⊂ I ^ biC then works for the conclusion of the lemma. Now we have the theorem. Theorem 2.4.2. Let T be a 1-based supersimple theory. Let C be a set. Suppose a is an imaginary element and b is an arbitrary imaginary element with SU(b/C) < ω. If a 6 | b then there is a stable formula in tp(a/bC) which forks over C. ^ C Proof. This proof closely follows the proof of Theorem 2.3.4. Suppose a 6 | b. ^ C We proceed by induction on SU(b/C). If SU(b/C) = 1 then Proposition 2.3.1 produces a stable forking formula in tp(a/bC). Now suppose SU(b/C) = α and the proposition holds for all elements having smaller rank (over C). Choose a forking formula ϕ(x; bc) ∈ tp(a/bC) (for some c ∈ C), and a sequence I = (bic)i∈ω containing b which is indiscernible over aC and for which a 6∈ acl(IC) and a | I; ^ bC such a sequence exists by Lemma 2.4.1. Let e = Cb(a/IC). e ∈ bdd(bC) since a | IC. In fact, the above rank argument could be repeated with any b0 ∈ I to ^ bC give e ∈ bdd(b0C). However, b∈ / acl(e), since b ∈ acl(e) would imply b ∈ acl(b0C) for any b0 ∈ I, which is impossible since b and b0 are both in the same indiscernible sequence I. Hence SU(b/e) > 0 and SU(e/b) = 0. The Lascar inequalities then show SU(e) < SU(b). a 6 | I, and a | I imply a 6 | e. DWIP produces a ^ C ^ e ^ C finite, real tuple d ⊂ e with a 6 | d. Note that SU(d) ≤ SU(e) and d ∈ acl(bC). ^ C Since SU(dC) < SU(bC), apply the induction hypothesis to get a stable forking formula inside tp(a/eCˆ ). Then Corollary 2.3.3 gives a stable forking formula in tp(a/bC). 37 CHAPTER 3 SEQUENCE EXTENSIONS This chapter considers a property on indiscernible sequences. In a stable the- ory, a pair of indiscernible sequences having a common extension is an equivalence relation on sequences. A natural question is whether this condition is peculiar to stable theories, or whether it holds for other theories. Since indiscernible sequences in stable theories are related to canonical bases, it seems natural to consider sim- ple theories, for which the notion of non-forking independence makes sense and types over boundedly closed sets have canonical bases. The question is, what would cause two sequences which “should” have a common extension to fail to have one. We formalize the question by defining a Y-formation and ask whether a Y-formation has a common extension. We show that this is certainly not true of any non-simple theory. We also provide some sufficient conditions for a theory to have common extensions. Whether every simple theory has common extensions is left as an open question. The idea of characterizing theories by properties on indiscernible sequences is not new. In [18], Shelah shows a theory is stable if and only if every indis- cernible sequence is an indiscernible set. More recently, Scow has characterized NIP theories as exactly the ones where every ordered graph indiscernible sequence is indiscernible, up to some technical conditions (see [16]). 38 Unlike the other chapters, this chapter does not begin with assumptions on the simplicity of T . 3.1 Common Extensions Given an indiscernible sequences I and J, we say J extends I and write I_J if the sequence formed by their concatenation (with I before J) is an indiscernible sequence. For every sequence I and every pair J, K of extensions to I, the moti- vating question is whether there is a sequence extending both I_J and I_K. Definition 3.1.1. Say a tuple (I, J, K) of indiscernible sequences is a Y-formation if both concatenations I_J and I_K form indiscernible sequences. A Y-formation (I, J, K) has a common extension if there is an indiscernible sequence L such that I_J _L and I_K_L are both indiscernible; we call L the common extension.A theory T has Y-extensions if every Y-formation has a common extension. There is a partial type Γ0(X; YZ) on infinite tuples of variables X = (x0, x1,...) (same with Y and Z) which says Y _Z_X is indiscernible. If I_J is an in- discernible sequence then Γ0(X; IJ) is always consistent by Compactness. The partial type Γ given by the union Γ(X; YZW ) = Γ0(X; YZ) ∪ Γ0(X; YW ) is consistent if and only if (Y,Z,W ) has a common extension. This shows having a common extension is type-definable. Some theories do have Y-extensions, so the concept is not vacuous, as we shall see. Proposition 3.1.2. If T is stable then T has Y-extensions. 39 Proof. Let (I, J, K) be a Y-formation; we need to show it has a common extension. Since I is indiscernible over the set JK, we can find a sequence H such that both H_I_J and H_I_K are indiscernible. Since T is stable, indiscernible sequences are also indiscernible sets. Hence I_J _H and I_K_H are both indiscernible sequences, and so H is a common extension. Conversely, here are two examples of theories which do not have Y-extensions. Example 3.1.3. Let T = Th(Q, <) be the theory of dense linear orders without endpoints. Choose a sequence I = (aibi : i < ω) giving endpoints of a nested sequence of intervals That is, ai < ai+1 < bi+1 < bi for all i < ω, and so in- terval [ai+1, bi+1] ⊂ [ai, bi] for all i < ω. This sequence is indiscernible over ∅. Compactness and saturation of M give two disjoint intervals [c, d], [e, f] inside the intersection ∩i[ai, bi]. Let J = (cidi : i < ω) be a sequence of nested intervals with c0d0 = cd, and K = (eifi : i < ω) be a sequence of nested intervals with e0f0 = ef. Then (I, J, K) is a Y-formation with no common extension since by indiscernibility any such extension must lie inside the interval [c, d] ∩ [e, f] = ∅, which is impossible. Example 3.1.4. Let T = Tfeq be the theory of generic parameterized equivalence relations. Let M be a two sorted structure U,V having a single ternary relation R ⊂ U × U × V . We think of the elements in sort U as points and elements of V as names of equivalence relations, in the sense that for any e ∈ V , the projection R(x, y, e) ⊂ U ×U is an equivalence relation. T is the model companion of Th(M). Let I = (aiei : i < ω) be an indiscernible sequence where ai is a point in the first sort, ei is an equivalence relation in the second sort, and for all j > i, R(ai, aj, ei) holds. Let J = (bifi : i < ω) be an indiscernible sequence extending 40 I. Since the equivalence relations are generic with respect to each other, there is 0 0 some element b0 ≡If0 b0 in a different f0-class than b0, that is ¬R(b0, b0, f0) holds. 0 Let K be an If0-conjugate of J containing b0f0. Then (I, J, K) is a Y-formation with no common extension. Suppose there were a common extension L; choose an 0 element cg ∈ L. By indiscernibility R(b0, c, f0) and R(b0, c, f0) both hold. Since 0 R(−, −, f0) is an equivalence relation, R(b0, b0, f0) holds, contradiction our choice 0 of b0. Hence there is no common extension. The last example highlights that the sequences J and K may have any relation whatsoever to each other; in the example they both contain the element f0. The examples presented are also both canonical examples of theories with the tree property, the first one has the tree property of the first kind, and the second has the tree property of the second kind. This suggests that the tree property causes problems. This is, in fact, what is happening. As stated earlier, the tree property can be thought of as “dividing gone bad.” One manifestation of this slogan is the following (paraphrased from Wagner [20]). Fact 3.1.5. [20, Lemma 2.3.7] Let ϕ(x; y) have the k-tree property Then for every linearly ordered index set I there is an indiscernible sequence (biai : i ∈ I) such that |= ϕ(bi; a0) and ϕ(x; ai) k-divides over {bjaj : j < i} for all i ∈ I. Having an indiscernible sequence where each element divides over its prede- cessors allows us to assemble a non-extending Y-formation. Lemma 3.1.6. If ψ(x; y) has the tree property then there is a Y-formation which does not have a common extension. Proof. Suppose ψ(x; y) has the tree property. We may assume it has 2-tree prop- erty, so by Fact 3.1.5 there is a sequence (biai : i < ω + ω) such that |= ψ(bi; aj) iff 41 i ≥ j and ψ(x; aj) 2-divides over (bkak : k < j) for all j. Let I = (biai : i < ω) be the first half of the sequence and J = (biai : ω ≤ i < ω + ω) be the second half. j Since ψ(x; aω) 2-divides over I, there is an I-indiscernible sequence (aω : j < ω) 0 j with aω = aω such that {ψ(x; aω): j < ω} is 2-inconsistent. Let σ be an I- 0 1 0 automorphism taking aω to aω. Let J = σ(J) be the indiscernible sequence which is the image of J under σ. We will now show (I, J, J 0) is a Y-formation which does not have a common extension. have a common extension claim: (I, J, J 0) is _ 0 a Y-formation. By construction I J is an indiscernible sequence. Since J ≡I J, the extension I_J 0 is also. claim: (I, J, J 0) has no common extension. Suppose it did and L were a com- mon extension. Then both I_J _L and I_J 0_L are indiscernible sequences. Let 0 1 0 ba be an element from L. Recall aω ∈ J and aω ∈ J . By indiscernibility and 0 1 0 1 construction we have both |= ψ(b, aω) and |= ψ(b, aω). But ψ(x, aω) ∧ ψ(x, aω) is inconsistent, contradicting the fact that b is a common witness. Thus there is no common extension. Corollary 3.1.7. If T has Y-extensions then T is simple. Proof. If T is not simple, some formula has the tree property. Hence, by Lemma 3.1.6 there is a Y-formation with no common extension. Now we investigate the converse, that T being simple implies T has Y-extensions. In the proof of Lemma 3.1.6 dividing provided an infinite sequence which was 2- inconsistent, yet we only used the first two elements of the sequence. We capture this by introducing an a priori weaker condition called the (2, 2)-small tree prop- erty. With an essentially identical proof as Lemma 3.1.6 we show that if T has no formula with the (2, 2)-small tree property, then T has Y-extensions. This is done 42 in §3.3. For the rest of this section we will assume T is simple. The type amalga- mation theorem for simple theories gives a partial converse to Corollary 3.1.7. Proposition 3.1.8. Let T be a simple theory. Suppose (I, J, K) is a Y-formation and J | K. Then (I, J, K) has a common extension. ^ I Proof. The type Γ0(X; I) saying X is an indiscernible extension of I has non- forking extensions Γ (X; IJ) and Γ (X; IK). Since J | K, the type Γ(X; IJK) = 0 0 ^ I Γ0(X; IJ) ∪ Γ0(X; IK) is consistent by the type amalgamation theorem. Hence (I, J, K) has a common extension. Fix a sequence I, and write J ∼I K if (I, J, K) has a common extension. This relation is symmetric and reflexive, so the transitive closure of ∼ is an equiva- lence relation ≈I on sequences extending I. In fact, ≈I is inside the two step 2 0 iterate ∼I , since for a Y-formation (I, J, K), we may find a sequence J ≡I J such that J 0 | JK. Then both (I, J, J 0) and (I,J 0,K) are Y-formations, and ^ I Proposition 3.1.8 tells us that both have common extensions, so J ≈I K. Hence, ≈I is type definable. In fact, this is a trivial relation since any pair of sequences which extend I as an indiscernible sequence are ≈I related. This is because this equivalence relation is intimately related to canonical bases. Turning now to Y-formations (I, J, K) which are not independent over I, it is not clear how to handle them. As a first step, we find consequences from the failure to have a common extension. Lemma 3.1.9. Let T be simple. If (I, J, K) has no common extension then the following must hold. 1. J 6 | K ^ I 43 2. There is a formula ϕ(¯x;y ¯) over I and tuples ¯b ⊂ J, c¯ ⊂ K such that ¯ ¯ ¯ ϕ(¯x; b) ∧ ϕ(¯x;c ¯) is inconsistent, b ≡I c¯, and ϕ(¯x; b) does not divide over I. Proof. Suppose the hypothesis. J 6 | K follows immediately from Lemma 3.1.8. ^ I Since there is no common extension, the type Γ(X; IJK) is inconsistent, and so there is a finite inconsistent subset ∆(¯x;a ¯¯bc¯) ⊂ Γ(X; IJK) witha ¯ ⊂ I, ¯b ⊂ J, ¯ ¯ andc ¯ ⊂ K. We may assume b ≡I c¯ since J ≡I K and by extending b orc ¯ as necessary. We may also assume there is some finite ∆0 ⊂ Γ0(X; IY ) such that ¯ ¯ ∆(¯x;a ¯bc¯) ⊆ ∆0(¯x;a ¯b) ∪ ∆0(¯x;a ¯c¯). Let ϕ(¯x;a ¯y¯) be the conjunction of formulas ¯ in ∆0. Then ϕ(¯x;a ¯b) ∧ ϕ(¯x;a ¯c¯) is inconsistent since it is implies ∆, and ∆ is ¯ inconsistent. Since Γ0(X; IJ) does not fork over I, and implies ϕ(¯x;a ¯b), it follows that ϕ(¯x;a ¯¯b) does not divide over I. 3.2 Graph Relation from No Common Extension This section expands on Lemma 3.1.9 by studying the formula ϕ which is an inconsistency witness to a Y-formation not having a common extension. The goal is to find necessary conditions on a theory which do not have common extensions. To this end, we derive an interpretable graph whose edge relation causes forking. In this way, this graph is fundamentally different from the random graph, and this suggests theories having no common extensions are much different form those which do. Definition 3.2.1. If (I, J, K) is a Y-formation which does not have a common extension, and ψ(x; ab), ψ(x; ac) are formulas as in Lemma 3.1.9 with a ∈ I, b ∈ J, c ∈ K such that ψ(x; ab) ∧ ψ(x; ac) is inconsistent, then we will say ψ and (a, b, c) 44 are an inconsistency witness to having a common extension (or an NCE witness for short). If ψ(x; ay) is an NCE witness then we can form a symmetric formula θψ(y; z) ≡ ¬∃x[ψ(x; ay) ∧ ψ(x; az)] over a. A graph relation on realizations of tp(b/I) comes from θ = θψ whereby a pair of tuples are connected by an edge iff θ holds between them. We will let Gθ denote this graph, which depends on the initial choice of ψ. For the rest of this section fix some ψ(x; b), and hence some θ and some Gθ. We will now just write G for the graph. Lemma 3.2.2. For some n < ω, G omits all n-complete subgraphs. Proof. Suppose not, so for every n we can find an n-complete subgraph. Then, by Compactness we can find an infinite subgraph D = (di : i < ω) which is complete, i.e. |= θ(di; dj) for all i < j < ω. By Compactness and Erd¨os-Rado, we may assume D forms an indiscernible sequence over I. In particular, the increasing two-types in the sequence are all θ connected, which from the definition of θ means the set {ψ(x; adi): i < ω} is pairwise inconsistent. Since d0 ≡I b, we have shown ψ(x; ab) divides over I. But ψ does not divide over I, by Lemma 3.1.9. The contradiction proves that G omits almost all finite complete subgraphs. Let n be the smallest integer such that G omits the complete subgraph Kn. Let q(¯y) = tp(b/I). Since the subgraph Kn is omitted, we have [ _ q(yi) ` ¬θ(yi; yj). i≤n i 45 Note that θ is negated, since the absence of an edge comes from the negation of θ. By Compactness, there is some formula ξ(y) ∈ q such that ^ _ ξ(yi) → ¬θ(yi; yj). (3.1) i≤n i The set H = ξ(M) is definable over I and θ provides a graph relation on H which ∗ omits Kn by the implication in equation 3.1. Let θ (y; z) ≡ ξ(y)∧ξ(z)∧θ(y; z) be the edge relation restricted to H. For any b ∈ H there is no indiscernible sequence over I for which θ∗ holds between distinct pairs of elements. In fact, θ∗ forks over I. Lemma 3.2.3. If b ∈ H then θ∗(y; b) forks over I. Proof. Choose arbitrary b ∈ H and suppose θ∗(y; b) does not fork over I. We will construct a sequence (bi : i < ω) of elements such that for all n, bn ≡I b, b | b ··· b , and |= θ∗(b , b ) for all i < n. This sequence will then contradict n ^ I 0 n−1 n i Lemma 3.2.2. Begin by putting b0 := b. Suppose we have b0 ··· bn−1, and we wish to find bn. Put p(y) = tp(b/I). Note that p is an amalgamation base since I can be viewed as an indiscernible sequence in tp(b). Let r(y; z) be a non-forking completion of p(y) ∪ p(z) ∪ {θ∗(y; z)}. The type r(y; bi) is consistent for each i < n since bi ≡I b. Each r(y; bi) is a non- forking extension of p(y), and since {b0, ··· , bn−1} are an independent set over I, 46 the Type Amalgamation Theorem gives a realization bn of the type [ r(y; bi) i The previous lemma provides a qualitative difference between the graph struc- ture on H and the random graph, or even the triangle-free random graph. Whereas in the latter graphs the edge relation does not cause forking, it does in graph H. In particular, since the edge relation forks over I, the usual argument showing the triangle-free random graph is not simple—that is, the Type Amalgamation Theorem—does not apply to H. It also means we cannot conclude T is not sim- ple by the fact that H omits all complete subgraphs of size ≥ n. 3.3 Small-Tree Property A weaker property than the tree property is the (k, m)-small tree property. For this we only require an m-branching tree of parameters of height ω, but otherwise identical conditions as the k-tree property. Definition 3.3.1. Say a formula φ(x; y) has the (k, m)-small tree property for <ω 2 ≤ k ≤ m if there is a m-branching tree of height ω of tuples {bs : s ∈ m } (each tuple has the same length as y), such that • Each path is consistent: for each path σ ∈ mω the set {φ(x; b ): i < ω} is σi consistent, and <ω • Immediate children are inconsistent: for each node s ∈ m , the set {φ(x; bs_i): i < m} is k-inconsistent. 47 The k-tree property is synonymous with the (k, ω)-small tree property. The next proposition gives some immediate properties of small-tree formulae. This property is not as nice to work with as the usual tree property since the finiteness of the branching limits the possible combinatorics. Proposition 3.3.2. 1. If ψ(x; y) has the (k, m)-small tree property then ψ also has the (k0, m0)-small tree property for all k0 ≥ k and m0 ≤ m (and k0 ≤ m0). 2. For a fixed k, ψ(x; y) has the (k, m)-small tree property for every m < ω if and only if ψ(x; y) has the k-tree property. 3. If ψ(x; y) has the (2, m)-small-tree property for some m ≥ 2, then ψ is unstable. <ω Proof. (1) Suppose ψ has the (k, m)-small tree property. Let {bs : s ∈ m } be parameters witnessing this. Fix some k0 ≥ k and m0 ≤ m. Then by pruning 0 ω 0 branches from the tree we get a tree {bs : s ∈ (m ) } which is m -branching, and for each node, the immediate children are k0 inconsistent. (2) One direction follows from part (1). The other direction is by Compactness. We can witness the k-inconsistency on any finite piece of ω<ω by an m-branching tree for some m < ω. <ω (3) Let ψ have the (2, m)-small tree property, and B = {bs : s ∈ m } be a set ω of parameters witnessing it. For each path η ∈ m let πη(x) = {ψ(x; bs): s ⊂ η} be the partial type formed from parameters lying on η. For any distinct η, τ ∈ mω, the type πη ∪ πτ is inconsistent since there is some smallest i ∈ ω such that η(i) 6= τ(i) hence ψ(x; bs_η(i))∧ψ(x; bs_τ(i)) is inconsistent where s = η (i−1) = τ (i − 1). The inconsistency comes from the immediate children of node s being 48 2-inconsistent. Thus there are 2ω ψ-types over a countable set, and so ψ is not stable. The third proof builds a tree that is similar to the “2-tree” construction which characterizes stability. The 2-tree construction builds continuum many types using both positive and negative instances of the formula. The (2,2)-small tree construc- tion uses only positive instances. The random graph provides a theory which is simple unstable, yet does not have the (2,2)-small tree property. It also shows that having the (2,2)-small tree property is strictly stronger than unstability. The small tree property is related to the D(p, ∆, k, λ) ranks found in [19] and [5]. In the same way the usual D rank measures the “degree” of the tree property present, for λ < ω the D(p, ∆, k, λ) rank measures the degree of the (k, λ)-small tree property present. Just as with the tree property, we can make the distinction between the small- tree property of the first and second kinds. In the latter case, ψ has the (k, m)- j small tree property of the second kind if there is an array of parameters (bi : i < ω j m, j < ω) such that for any path σ ∈ m the set {ψ(x; bσ(j)): j < ω} is consistent, <ω yet for each level, s ∈ m the next row {ψ(x; bs_i): i < m} is k-inconsistent. Alternatively, ψ has the (k, m)-small tree property of the first kind if there <ω is a witnessing tree of parameters (bs : s ∈ m such that that not only are the immediate children of any given node inconsistent, but if s, t ∈ m<ω are two incomparable nodes (that is, they do not lie on a common path σ ∈ mω), then ψ(x; bs) ∧ ψ(x; bt) is inconsistent. A formula ψ has the SOP2 property if there <ω is a tree (bs : s ∈ 2 ) of parameters such that for any two incomparable nodes <ω s, t ∈ 2 , the conjunction ψ(x; bs) ∧ ψ(x; bt) is inconsistent. A formula having the (k, m)-small tree property of the first kind has SOP2, almost by definition, 49 and a formula having SOP2 has the 2-tree property, so it has the (2, m)-small tree property of the first kind for all m < ω. Proposition 3.3.3. For a formula ψ the following are equivalent. 1. ψ has the (k, m)-tree property of the first kind for some k, m < ω 2. ψ has SOP2 We tie a theory not having common extension to the existence of a formula with the (2,2)-small tree property. Lemma 3.3.4. If T does not have common extensions then there is a formula with the (2,2)-small tree property. Proof. Suppose T does not have common extensions, so there is a Y-formation (I, J, K) witnessing it. By Compactness we may assume I, J, and K have count- able, dense order type. Let ψ(x; yz), and finite tuples a ⊂ I, b ⊂ J, and c ⊂ K such that b ≡I c and ψ(x; ba) ∧ ψ(x; ca) is inconsistent, as provided by Lemma 3.1.9. Our goal is to construct a tree in tp(b/I) which witnesses that ψ has the (2,2)-small tree property. By Compactness, again, we may assume there is a countable dense sequence I0 such that (I_I0, J, K) is a Y-formation. Since I0 and J have countable dense order _ 0_ 0 0_ type, and I I J is indiscernible, I ≡I I J. Hence there is an automorphism 0 0_ σ fixing I and taking I to I J. Let b00 := σ(b) and b01 := σ(c). Likewise 0 0_ there is an automorphism τ fixing I and taking I to I K. Put b10 := τ(b) and b11 := τ(c). <ω <ω Continue in this way to define (bs : s ∈ 2 ). The tree (bsa : s ∈ 2 ) then witnesses that ψ has the (2,2)-small tree property. 50 <ω The tree (bs : s ∈ 2 ) constructed in the proof of Lemma 3.3.4 is such that <ω at each level of the tree s ∈ 2 , the children bs_0 and bs_1 have the same type over their common initial segment Bvs = {bt : t v s} (here v is the relation of being an initial segment).1 Definition 3.3.5. Say ψ has the strong (k, m)-small tree property if ψ has the <ω (k, m)-small tree property and there is a witnessing tree (bs : s ∈ m ) with the following additional properties <ω 1. For each s ∈ m the immediate children have the same type over Bvs: that _ _ is, bs i ≡Bvs bs j for all i, j < m. <ω 2. If s ∈ m and t v s is an initial segment then |= ψ(bs; bt). We restate Lemma 3.3.4 in the new language. Corollary 3.3.6. If T does not have common extensions then there is a formula with the strong (2,2)-small tree property. A formula having the strong (2,2)-small tree property is equivalent to there being a Y -formation with no common extension. Lemma 3.3.7. If ψ(x; y) has the strong (2,2)-small tree property then there is an indiscernible sequence (bici : i < ω) such that 1. bn ≡(b 2. |= ψ(bj; bi) iff j > i 3. ψ(x; bi) ∧ ψ(x; ci) is inconsistent for all i. 1This change says in part that the children of each node 2-quasi-divide over their initial segment (cf. [3]). 51 <ω Proof. Given a witnessing tree D = (ds : s ∈ 2 ), put bi = d0(i)_0 and ci = d0(i)_1, where the function 0(k) maps k 7→ 000 ··· 0 (with k 0’s). By Compactness, we extend the sequence (bici : i < ω) to a sequence L of length λ. The sequence L satisfies conditions (1), (2) and (3). We may now extract the required indiscernible sequence. Lemma 3.3.8. Let T be simple. The following are equivalent. 1. There is a formula ψ having the strong (2,2)-small tree property. 2. T does not have common extensions. Proof. (1) ⇒ (2) Use Lemma 3.3.7 to get an indiscernible sequence I = (bici : i < _ ω). Let J = (bici : ω ≤ i < ω + ω) be an extension of I so I J is indiscernible. 0 0 Then let I = (bi : i < ω) and J = (bi : ω ≤ i ≤ ω + ω) be projections of I 0_ 0 and J to the first tuple. The sequence I J is still indiscernible. Since bω ≡I cω, 0 0 let K be the conjugate of J under an automorphism fixing I and taking bω 0 0 0 to cω. Then (I ,J ,K ) form a Y-formation which has no common extension since ψ(x; bω) ∧ ψ(x; cω) must be satisified by any common extension, yet it is inconsistent by construction. (2) ⇒ (1) Follows from Lemma 3.3.6. 3.4 Fans In this section we count the number of possible non-extending classes over an indiscernible sequence I. This turns out to be bounded in simple theories. Definition 3.4.1. Given an indiscernible sequence I, we say a set J of indis- cernible sequences is a fan over I if for each J ∈ J , I_J is indiscernible. 52 Say J is an indiscernible fan over I if in addition to being a fan over I there is some linear ordering of J making J into an indiscernible sequence (of sequences) over I. We say a fan J over I is pairwise non-extending (PNE) if for any pair of distinct J, J 0 ∈ J ,(I, J, J 0) does not have a common extension. The boundedness comes from the fact that simple theories do not admit in- discernible PNE fans over I. We show that the existence of an indiscernible PNE fan allows one to find a formula with the tree property. Lemma 3.4.2. Suppose T is a simple theory. Let I be a sequence and J = {Ji : i < ω} be an indiscernible fan over I. Then J is not pairwise non-extending. _ _ Proof. The type Γ0(X; IY ) stating I Y X is an indiscernible sequence does not fork over I since it is finitely satisfiable in I. We consider the sequence J as S a sequence of parameters Y for Γ0(X; IY ). The set Σ(X) = i<ω Γ0(X; IJi) is consistent since Γ0(X; IJ0) does not fork over I, and J is indiscernible over I. Let L |= Σ. Then L is simultaneously a common extension of each Ji, hence every pair of elements Ji,Jk have a common extension, and J is not pairwise non-extending. The proceeding proof requires the fan to be indiscernible over I to invoke lemma showing Σ is consistent. The indiscernibility ensures for each pair of se- quences Ji,Jk, the type Γ(X; IJiJk) is inconsistent via the same formula ψ. Then the conjugates of the parameters to ψ form an indiscernible sequence witnessing the 2-dividing of ψ over I, but since ψ ∈ Γ0, it cannot divide over I. Corollary 3.4.3. Let T be simple. Let I and J be sufficiently long fan. Then there are some distinct pair J, J 0 ∈ J such that (I, J, J 0) has a common extension. 53 Proof. Suppose no such pair from J has a common extension, making J PNE. Since J is sufficiently long, there is some indiscernible fan K over I whose EM- types come from tuples of sequences in J . In particular, K is PNE. But, by Lemma 3.4.2 K is not PNE contradiction our initial assumption. Hence J is not PNE, and some pair must have a common extension. Corollary 3.4.4. Let T be a simple theory. There is a bound λfan such that if (I, J ) is a PNE fan then |J | < λfan. This bound comes from the Erd¨os-Rado bound to extract an indiscernible sequence from J . We define a combinatorial condition on fans, the existence of which is equiva- lent to the existence of pairwise-non-extending fans. Definition 3.4.5. We say a fan I, J is wide if for every N < ω there is some i < ω such that for every k < N,(I,Ji,Jk) does not have a common extension. The reason for defining wide this way is for its negation: if the fan I, J is not wide then there is some N such that for every i < ω there is some k < N such that (I,Ji,Jk) has a common extension. We will show that the existence of wide fans is equivalent to the existence of PNE fans. Clearly, any fan which is pairwise non-extending is wide, since no pair of sequences have a common extension. The next lemma shows that any wide fan contains an infinite subset which is pairwise non-extending. So the two concepts are essentially the same for our purposes. Lemma 3.4.6. Let T be any theory. Suppose I and J = {Ji : i < ω} form a fan. If J is wide then there is an infinite subset K of J such that (I, K) is a PNE fan. 54 Proof. From wideness we get a function f : ω → ω such that given N, for every k < N,(I,Jk,Jf(N)) does not have a common extension. It must be the case that f(n) ≥ n for every n. Otherwise, if f(n) < n we would have (I,Jf(n),Jf(n)) does not have a common extension, but it does by Compactness. Let `0 = 0 and `n = f(`n−1 + 1). The sequence (`i : i < ω) form an infinite, increasing sequence of natural numbers. Let K = {J`i : i < ω} be our infinite subset of J . We need to show that no pair of elements in K has a common 0 0 extension over I. To this end, choose K,K ∈ K. Then K = J`i and K = J`j for some i, j by construction. We may assume i < j so `i < `j. By definition `j = f(`j−1+1) so (I,Jk,J`j ) does not have a common extension for all k < `j−1+1. 0 In particular, `i ≤ `j−1 < `j−1 + 1 so (I,J`i ,J`j ) = (I,K,K ) does not have a common extension. Lemma 3.4.7. Let T be any theory. The following are equivalent. 1. T has PNE fans 2. T has wide fans Proof. Combine Lemma 3.4.6 with the observation that any PNE fan is wide. 3.5 A Small Tree Example This section presents a simple theory which has the (2,2)-small tree property. The theory is from Kim [8], and is presented as Example 2.7.11 in Wagner [20]. The language consists of two symbols L = {R,Q} where R is a binary relation and Q is a unary predicate. We can interpret the theory in the random bipartite graph like so. Let M is a model of the bipartite graph with edge relation G and 55 predicate P naming one of the classes, we choose two distinct elements a, b ∈ M and define two sets: D = {(x, w): x ∈ P, w ∈ {a, b}} and E = {y : y∈ / P }. Define a bipartite edge relation R ⊂ D × E by G(x, y) if w = a R((x, w), y) ⇐⇒ ¬G(x, y) if w = b Then T is simply the theory of M in language L with the interpretation of R as given and with Q naming the set D. The theory T is essentially the bipartite ran- dom graph, except each vertex v in Q has a unique corresponding “mirror” vertex that is connected to exactly the elements in ¬Q no connected to v. This theory is simple since it is interpretable in the simple theory of the random bipartite graph. We show the formula ψ(x; y) ≡ R(y, x) has the (2,2)-small tree property by building a witnessing tree. If c ∈ Q, let c0 represent its mirror element. Choose a countable subset V of Q which is “mirror-free”, that is for any c ∈ V , the element 0 <ω c ∈/ V . Fix a bijection f : 2 → V . Define cs by _ bf(s) if ∃t[s = t 0] cs = 0 _ bf(t_0) if ∃t[s = t 1] <ω (So only “half” the image of f is used). We claim the tree (cs : s ∈ 2 ) witnesses the (2,2)-small tree property of ψ. For each path σ ∈ 2ω, the set W = {c : σ σi i < ω} is mirror free since it comes from a subset of V by either mirroring or not mirroring each element. Hence {ψ(x; c): c ∈ Wσ} is consistent. For each node cs, <ω for s ∈ 2 , the immediate children are cs_0 and cs_1, which by construction are 0 bf(s_0) and bf(s_0), which are mirror to each other. Hence ψ(x; cs_0) ∧ ψ(x; cs_1) 56 is inconsistent. Thus ψ has the (2,2)-small tree property. However, we cannot construct a Y-formation with no common extension from this tree. Let us try. Begin with a sequence I = (aibi : i < ω) where ai ∈ Q and bi ∈ ¬Q for all i and R(ai, bj) holds iff i ≤ j. Let aωbω be a tuple extending the 0 0 sequence. Since aω is in Q, there is a mirror point aω. However, aω 6≡I aω since 0 aω is not R connected to any bj for j < ω, yet aω is R connected to every bj. The difference in types prevents us from constructing a non-extending Y-formation in the usual way by forming a sequence J containing aωbω and then using an 0 I-automorphism f to take aω to aω and putting K = f(J). 57 BIBLIOGRAPHY [1] Hans Adler. Explanation of Independence. PhD thesis, Universit¨atFreiburg, 2005. [2] Enrique Casanovas. 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