OON MODEL-THEORETIC TREE PROPERTIES
ARTEM CHERNIKOV AND NICHOLAS RAMSEY
Abstract.
We study model theoretic tree properties (TP , TP , TP ) andtheir associated cardinal invariants ( κ cdt , κ sct , κ inp , respectively). In particu-lar, we obtain a quantitative refinement of Shelah’s theorem (TP ⇒ TP ∨ TP )for countable theories, show that TP is always witnessed by a formula in a sin-gle variable (partially answering a question of Shelah) and that weak k − TP is equivalent to TP (answering a question of Kim and Kim). Besides, we givea characterization of NSOP via a version of independent amalgamation oftypes and apply this criterion to verify that some examples in the literatureare indeed NSOP . Contents
1. Introduction 12. Preliminaries on indiscernible trees 33. Cardinal invariants and tree properties 64. TP and weak k − TP theories 166. Examples of NSOP theories 207. Lemmas on preservation of indiscernibility 29References 351. Introduction
One of the central tasks of abstract model theory is to understand what kindsof complete first-order theories there are and how complicated they can be. Inpractice, this is achieved by classifying theories according to the combinatorialconfigurations that do or do not appear among the definable sets in their models.The most meaningful of these configurations, the so-called dividing lines , have theproperty that their absence signals the existence of some positive structure, whiletheir presence indicates some kind of complexity. Dividing lines come in two flavors:local properties, which describe the combinatorics of sets defined by instances of asingle formula, and global properties, which describe the interaction of definable setsgenerally. Stability, simplicity, NIP are examples of the former, while ω -stability,supersimplicity, and strong dependence are examples of the latter (see e.g. [10]). Date : October 24, 2016.A.C. was partially supported by ValCoMo (ANR-13-BS01-0006), by EPSRC grantEP/K020692/1, by the Fondation Sciences Mathematiques de Paris (FSMP) and by a publicgrant overseen by the French National Research Agency (ANR) as part of the Investissementsd’avenir program (reference: ANR-10-LABX-0098). a r X i v : . [ m a t h . L O ] O c t ARTEM CHERNIKOV AND NICHOLAS RAMSEY
In this paper, we study some questions around Shelah’s tree property TP andits relatives SOP , TP , TP and weak k -TP , as well as their global analoguesdetected by the cardinal invariants κ cdt ( T ), κ inp ( T ), and κ sct ( T ). Our point ofdeparture is the third chapter of Shelah’s Classification Theory . There, Shelah in-vestigates the global combinatorics of stable theories in terms of a cardinal invariant κ ( T ) quantifying the complexity of forking in models of T . In the final section ofthis chapter, he introduces variations on κ ( T ) with the invariants κ cdt ( T ), κ sct ( T ),and κ inp ( T ) and proves several results about how they relate. In contemporary lan-guage, these invariants bound the size of approximations to the tree property, thetree property of first kind, and the tree property of the second kind consistent with T , respectively. Later as the theory developed, a property of stable theories thatforking satisfies local character was isolated and theories satisfying this condition,the simple theories , were intensively studied [4, 21, 25]. These theories are exactlythe theories without the tree property, which is to say those theories with κ cdt ( T )bounded. Nonetheless, until recently, the aforementioned invariants have receivedvery little attention and many basic questions remain unaddressed.Here, we focus on two such questions. Shelah proved that a theory has the treeproperty if and only if it has the tree property of the first kind or the tree propertyof the second kind [20]. In terms of the invariants, this amounts to the assertionthat κ cdt ( T ) = ∞ if and only if κ inp ( T ) + κ sct ( T ) = ∞ . It is natural to ask ifthis relationship persists when κ cdt ( T ) is bounded — in other words, if the equality κ cdt ( T ) = κ inp ( T ) + κ sct ( T ) holds in general. Shelah also proved that κ cdt ( T ) = κ is always witnessed by a sequence of formulas in a single free variable when κ isan infinite cardinal or ∞ . Recently, the first named author proved an analogousresult for κ inp ( T ) [8]. We consider here whether or not the computation of κ sct ( T )similarly reduces to a single free variable. These questions were both raised byShelah (Question 7.14 in [20]).We do not give a complete answer to any of them, but for each of these questionsthere are two model-theoretically natural special cases to consider: first, the caseof countable theories and, secondly, the case where one or more of the invariantsin question are unbounded (which reduces to a question about configurations ina single formula). In Section 3, we show that κ cdt ( T ) = κ inp ( T ) + κ sct ( T ) forcountable T . In Section 4, we show that if κ sct ( T ) = ∞ then this will be witnessedby a formula in a single free variable by showing that TP is always witnessedby a formula in one free variable. The main ingredient in our argument is thenotion of a strongly indiscernible tree , which is more easily manipulated than the s -indiscernible trees used in other studies of the tree property of the first kind.At the present state of the theory, the class of non-simple theories without thestrict order property is poorly understood even at the level of syntax. In their studyof the order (cid:69) ∗ , Dzamonja and Shelah introduced a weakening of TP called SOP [11]. Subsequently, Kim and Kim introduced two infinite families of propertiescalled k -TP and weak k -TP for k ≥ ⇐⇒ k -TP ⇐⇒ weak 2-TP = ⇒ weak 3-TP = ⇒ . . . = ⇒ SOP It was left open whether the properties weak k -TP are inequivalent for distinct k and whether or not weak k -TP is equivalent to TP [16]. In our work on provingthat TP is witnessed by a formula in one free variable, we obtained unexpectedlya simple and direct proof that the weak k -TP hierarchy collapses and that theyare all equivalent to TP . N MODEL-THEORETIC TREE PROPERTIES 3
In the final two sections of the paper, we study theories without the propertySOP . We show that independent amalgamation fails in a strong way in theo-ries with SOP and that they are in fact characterized by this feature. This givesrise to a useful criterion for showing that a theory is NSOP (and hence NTP ).Leveraging work of Granger [12] and Chatzidakis [6], this allows us to concludethat both the two sorted theory of infinite-dimensional vector spaces over an alge-braically closed field with a generic bilinear form, as well as the theory of ω -freePAC fields of characteristic zero are NSOP . Finally, we generalize the constructionof the theory of parametrized equivalence relations T ∗ feq to give a general methodfor constructing NSOP theories from simple ones. We learned after this work wascompleted that essentially the same construction had been studied by Baudisch [3],but our emphasis is different. We show that the independence theorem holds forthese structures, allowing us to obtain a proof that T ∗ feq is NSOP as a corollary. Acknowledgements.
We would like to thank the referee for numerous suggestionson improving the presentation, Zo´e Chatzidakis for her help with Lemma 6.7, andAlex Kruckman for pointing out an error and a way to fix it in Section 6.3 of anearlier version of the article.2.
Preliminaries on indiscernible trees
We fix a complete first-order theory T in a language L , M | = T is a monstermodel. In several of the arguments below, we will make use of the notion of anindiscernible tree. For our purposes, there are two different languages we will needto place on the index model: L s,λ = { (cid:67) , ∧ , < lex , ( P α : α < λ ) } and L = { (cid:67) , ∧ , < lex } where λ is a cardinal. We may view the tree κ <λ as an L s,λ − or L -structure in anatural way, interpreting (cid:67) as the tree partial order, ∧ as the binary meet function, < lex as the lexicographic order, and P α as a predicate which identifies the α th level(we will only consider κ = 2 and κ = ω ). See [17] and [23] for more details. Definition 2.1.
Suppose that ( a η ) η ∈ κ <λ and ( a α,i ) α<κ,i<ω are collections of tuplesand C is a set of parameters in some model.(1) We say ( a η ) η ∈ κ <λ is an s -indiscernible tree over C ifqftp L s,λ ( η , . . . , η n − ) = qftp L s,λ ( ν , . . . , ν n − )implies tp( a η , . . . , a η n − /C ) = tp( a ν , . . . , a ν n − /C ), for all n ∈ ω .(2) We say ( a η ) η ∈ κ <λ is a strongly indiscernible tree over C ifqftp L ( η , . . . , η n − ) = qftp L ( ν , . . . , ν n − )implies tp( a η , . . . , a η n − /C ) = tp( a ν , . . . , a ν n − /C ), for all n ∈ ω .(3) We say ( a α,i ) α<κ,i<λ is a mutually indiscernible array over C if, for all α <κ , ( a α,i ) i<λ is a sequence indiscernible over C ∪{ a β,j : β < κ, β (cid:54) = α, j < λ } . Lemma 2.2.
Let ( a η : η ∈ κ <λ ) be a tree strongly indiscernible over a set ofparameters C .(1) All paths have the same type over C : for any η, ν ∈ κ λ , tp(( a η | α : α <λ ) /C ) = tp(( a ν | α : α < λ ) /C ) .(2) For any η ⊥ ν ∈ κ <λ and any ξ , tp( a η , a ν /C ) = tp( a ξ(cid:95) , a ξ(cid:95) /C ) .(3) The tree ( a (cid:95)η : η ∈ κ <λ ) is strongly indiscernible over a ∅ C . ARTEM CHERNIKOV AND NICHOLAS RAMSEY
Proof. (1) This follows by strong indiscernibility of the tree as for any η, ν ∈ κ <λ ,qftp L (( η | α : α < λ )) = qftp L (( ν | α : α < λ )).(2) Let η ⊥ ν ∈ κ <λ be given, without loss of generality η < lex ν and let µ = η ∧ ν . Then there are i < j < κ so that µ (cid:95) (cid:104) i (cid:105) (cid:69) η and µ (cid:95) (cid:104) j (cid:105) (cid:69) ν . Thenqftp L ( η, ν ) = qftp L ( µ (cid:95) (cid:104) i (cid:105) , µ (cid:95) (cid:104) j (cid:105) ) = qftp L ( µ (cid:95) , µ (cid:95)
1) = qftp L ( ξ (cid:95) , ξ (cid:95) L (¯ η ) = qftp L (¯ ν ) implies qftp L (¯ η, ∅ ) = qftp L (¯ ν, ∅ ), provided ∅ is not enumerated in neither η nor ν . (cid:3) Lemma 2.3.
Let ( a η : η ∈ κ <λ ) be a tree s-indiscernible over a set of parameters C . (1) All paths have the same type over C : for any α, ν ∈ κ λ , tp(( a η | α ) α<λ /C ) =tp(( a ν | α ) α<λ /C ) .(2) Suppose { η α : α < γ } ⊆ κ <λ satisfies η α ⊥ η α (cid:48) whenever α (cid:54) = α (cid:48) . Then thearray ( b α,β ) α<γ,β<κ defined by b α,β = a η α (cid:95) (cid:104) β (cid:105) is mutually indiscernible over C .Proof. (1) This follows by s -indiscernibility of the tree as for any η, ν ∈ κ <λ ,qftp L s (( η | α : α < λ )) = qftp L s (( ν | α : α < λ )).(2) Fix α < γ and let A = { a η α (cid:48) (cid:95) (cid:104) β (cid:105) : α (cid:54) = α (cid:48) < γ, β < κ } ∪ C . As theelements of { η α : α < γ } are pairwise incomparable, it is easy to check that for any β < . . . < β n − < κ and β (cid:48) < . . . < β (cid:48) n − < κ ,qftp L s ( a η α (cid:95) (cid:104) β (cid:105) , . . . , a η α (cid:95) (cid:104) β n − (cid:105) /A ) = qftp L s ( a η α (cid:95) (cid:104) β (cid:48) (cid:105) , . . . , a η α (cid:95) (cid:104) β (cid:48) n − (cid:105) /A ) , which proves (2). (cid:3) Now we note that s -indiscernible and strongly indiscernible trees exist. Definition 2.4.
Suppose I is an L (cid:48) -structure, where L (cid:48) is some language. We saythat I -indexed indiscernibles have the modeling property if, given any ( a i : i ∈ I )from M , there is an I -indexed indiscernible ( b i : i ∈ I ) in M locally based on the( a i ): given any finite set of formulas ∆ from L and a finite tuple ( t , . . . , t n − ) from I , there is a tuple ( s , . . . , s n − ) from I so thatqftp L (cid:48) ( t , . . . , t n − ) = qftp L (cid:48) ( s , . . . , s n − )and also tp ∆ ( b t , . . . , b t n − ) = tp ∆ ( a s , . . . , a s n − ) . Fact 2.5. [17, 19, 23] Let I denote the L -structure ( ω <ω , (cid:69) , < lex , ∧ ) and I s bethe L s,ω -structure ( ω <ω , (cid:69) , < lex , ∧ , ( P α ) α<ω ) with all symbols being given their in-tended interpretations and each P α naming the elements of the tree at level α . Thenstrongly indiscernible trees ( I -indexed indiscernibles) and s -indiscernible trees ( I s -indexed indiscernibles) have the modeling property .In the arguments below, we will often argue by induction where at each stage itis necessary to modify a tree of tuples in a way that maintains the indiscernibilityof the tree. A convenient way of organizing these arguments is to make a catalogueof operations on indiscernible trees and prove that these operations preserve therelevant indiscernibility. N MODEL-THEORETIC TREE PROPERTIES 5
Definition 2.6.
Fix k ≥ k-fold widening of ( a η ) η ∈ ω <ω at level n is defined to be thetree ( a (cid:48) η ) η ∈ ω <ω where a (cid:48) η = a η if l ( η ) < n ( a ν(cid:95) ( ki ) (cid:95)ξ , . . . , a ν(cid:95) ( ki +( k − (cid:95)ξ ) if η = ν (cid:95) i (cid:95) ξ where ν ∈ ω n − , i ∈ ω, ξ ∈ ω <ω . (2) (stretching) The k-fold stretch of ( a η ) η ∈ ω <ω at level n is defined to be thetree ( a (cid:48)(cid:48) η ) η ∈ ω <ω where a (cid:48)(cid:48) η = a η if l ( η ) < n ( a η , a η(cid:95) , . . . , a η(cid:95) k − ) if l ( η ) = na ν(cid:95) k − (cid:95)ξ if η = ν (cid:95) ξ for ν ∈ ω n , ξ (cid:54) = ∅ (3) (fattening) Given a tree ( a η ) η ∈ <κ , define the k -fold fattening of ( a η ) η ∈ <κ to be the tree ( a ( k ) η ) η ∈ <κ by induction as follows: for each η ∈ <κ let a (0) η = a η . If ( a ( n ) η ) η ∈ <κ has been defined, for each η ∈ <κ , let a ( n +1) η =( a ( n )0 (cid:95)η , a ( n )1 (cid:95)η ). Let C k = { a η : η ∈ 1) + 1 defined by˜ η ( i ) = (cid:26) η ( i/k ) if k | i k -fold elongation of ( a η ) η ∈ κ <ω to be the tree ( b η ) η ∈ κ <ω where b η = ( a ˜ η , a ˜ η(cid:95) , . . . , a ˜ η(cid:95) k − ) . ARTEM CHERNIKOV AND NICHOLAS RAMSEY Proposition 2.7. (1) s -indiscernibility is preserved under widening, stretch-ing, fattening, restriction, and elongating.(2) Strong indiscernibility is preserved under restriction, fattening, and elon-gating. Moreover, if ( a η ) η ∈ <ω is strongly indiscernible, then the k -foldfattening ( a ( k ) ) η ∈ <ω is strongly indiscernible over C k .Proof. The proofs of these facts can be found in Section 7. (cid:3) Cardinal invariants and tree properties Definition 3.1. Suppose T is a complete theory and ϕ ( x ; y ) ∈ L is a formula inthe language of T .(1) ϕ ( x ; y ) has the tree property (TP) if there is k < ω and a tree of tuples( a η ) η ∈ ω <ω in M such that • for all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent, • for all η ∈ ω <ω , { ϕ ( x ; a η(cid:95) (cid:104) i (cid:105) ) : i < ω } is k -inconsistent.(2) ϕ ( x ; y ) has the tree property of the first kind (TP ) if there is a tree oftuples ( a η ) η ∈ ω <ω in M such that • for all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent, • for all η ⊥ ν in ω <ω , { ϕ ( x ; a η ) , ϕ ( x ; a ν ) } is inconsistent.(3) ϕ ( x ; y ) has the tree property of the second kind (TP ) if there is a k < ω and an array ( a α,i ) α<ω,i<ω in M such that • for all functions f : ω → ω , { ϕ ( x ; a α,f ( α ) ) : α < ω } is consistent, • for all α , { ϕ ( x ; a α,i ) : i < ω } is k -inconsistent.(4) T has one of the above properties if some formula does modulo T .It is easy to see that if a theory has the tree property of the first or second kind,then it also has the tree property. Remarkably, the converse is also true. Fact 3.2. [20] A complete theory T has TP if and only if it has TP or TP .The above theorem was first proven in different language, before any of thethree properties were actually defined. The purpose of this section is to prove arefinement of this theorem, by studying the relationship between approximationsto the tree property and those to the tree property of the first or second kind. In N MODEL-THEORETIC TREE PROPERTIES 7 order to do so, however, it will be necessary to return to the vocabulary in whichFact 3.2 was initially formulated. Definition 3.3. The following notions were introduced in [20].(1) A cdt -pattern of depth κ is a sequence of formulas ϕ i ( x ; y i ) ( i < κ, i successor)and numbers n i < ω , and a tree of tuples ( a η ) η ∈ ω <κ for which(a) p η = { ϕ i ( x ; a η | i ) : i successor , i < κ } is consistent for η ∈ ω κ ,(b) { ϕ i ( x ; a η(cid:95) (cid:104) α (cid:105) ) : α < ω, i = l ( η ) + 1 } is n i -inconsistent.A cdt-pattern with n i ≤ n for all i < κ , is called a (cdt , n )-pattern.(2) An inp -pattern of depth κ is a sequence of formulas ϕ i ( x ; y i ) ( i < κ ), se-quences ( a i,α : α < ω ), and numbers n i < ω such that(a) for any η ∈ ω κ , { ϕ i ( x ; a i,η ( i ) ) : i < κ } is consistent,(b) for any i < κ , { ϕ i ( x ; a i,α ) : α < ω } is n i -inconsistent.(3) An sct -pattern of depth κ is a sequence of formulas ϕ i ( x ; y i ) ( i < κ ) and atree of tuples ( a η ) η ∈ ω <κ such that(a) for every η ∈ ω κ , { ϕ α ( x ; a η | α ) : 0 < α < κ, α successor } is consistent,(b) If η ∈ ω α , ν ∈ ω β , α, β are successors, and ν ⊥ η then the formulas { ϕ α ( x ; a η ) , ϕ β ( x ; a ν ) } are inconsistent.If instead of (b), we have: for any pairwise incomparable ( η i : i < k ), { ϕ l ( η i ) ( x ; a η i ) : i < k } is inconsistent, then we call this a (sct , k )-pattern.(4) For X ∈ { cdt , sct , inp } , we define κ nX ( T ) to be the first cardinal κ so thatthere is no X -pattern of depth κ in n free variables, and ∞ if no such κ exists. We define κ X ( T ) = sup n ∈ ω { κ nX } . Remark . We note that the notion of a (cdt , n )-pattern strengthens that of acdt-pattern by imposing a uniform finite bound on the size of the inconsistency ateach level, while the notion of an (sct , n )-pattern weakens that of an sct-pattern byonly requiring any n incomparable elements to be inconsistent rather than any 2.One can regard an (sct , n )-pattern as an approximation to a witness to n -TP (seeDefinition 4.1 below). Observation 3.5. Fix a complete theory T .(1) κ n sct ( T ) ≥ n , κ n inp ( T ) ≥ n and κ n cdt ( T ) ≥ n for all n .(2) (a) κ cdt ( T ) = ∞ if and only if κ cdt ( T ) > | T | + if and only if T has TP.(b) κ sct ( T ) = ∞ if and only if κ sct ( T ) > | T | + and only if T has TP .(c) κ inp ( T ) = ∞ if and only if κ inp ( T ) > | T | + if and only if T has TP .(3) max { κ n sct ( T ) , κ n inp ( T ) } ≤ κ n cdt ( T ). Proof. (1) follows from the fact that “=” is in the language.(2) As each case is entirely similar, we’ll sketch the argument for (a) only. If κ cdt ( T ) > | T | + , then in the pattern witnessing it we may assume that ϕ i ( x, y i ) = ϕ ( x, y ) and k i = k , because | T | ≥ ℵ . This is a witness to TP. And then usingcompactness we can find a pattern witnessing that κ n cdt ( T ) > κ for any cardinal κ .(3) If ϕ i ( x ; y i ) ( i < κ ), ( a i,α : α < ω ), ( n i ) i<ω form an inp-pattern of depth κ , obtain a cdt-pattern of depth κ with respect to the same formulas by defining( b η ) η ∈ ω <κ by b η = a l ( η ) ,η ( l ( η ) − . (cid:3) Lemma 3.6. (1) If there is an sct-pattern (cdt-pattern) of depth κ modulo T , thenthere is an sct-pattern (cdt-pattern) ϕ α ( x ; y α ) , ( a η ) η ∈ ω <κ in the same number offree variables so that ( a η ) η ∈ ω <κ is an s -indiscernible tree. ARTEM CHERNIKOV AND NICHOLAS RAMSEY (2) If there is an inp-pattern of depth κ modulo T , then there is an inp-pattern ϕ α ( x ; y α ) ( α < κ ) , ( k α ) α<κ , ( a α,i ) α<κ,i<ω in the same number of free variables sothat ( a α,i ) α<κ,i<ω is a mutually indiscernible array.Proof. (1) By compactness and Fact 2.5.(2) This is Lemma 2.2 of [8]. (cid:3) Now we fix a complete theory T and for X ∈ { cdt , sct , inp } , we write κ X for κ X ( T ). Proposition 3.7. Assume that κ n cdt ≥ ℵ . Then either κ n inp ≥ ℵ or κ n sct ,k ≥ ℵ for some k ∈ ω (i.e. there are ( κ sct , k ) -patterns in n variables of arbitrary finitedepth). In fact, if κ n inp < ℵ , then one can take k = κ n inp .Proof. If κ n inp ≥ ℵ does not hold, then in fact we have κ n inp ≤ k for some k ∈ ω .Fix an arbitrary m ∈ ω , then by assumption and Lemma 3.6 we can find (cid:0) a η : η ∈ ω < m (cid:1) , ( ϕ i ( x, y i ) : i < m ) , ( k i : i < m ) an s -indiscernible cdt-patternwith | x | = n , i.e.:(1) (cid:0) a η : η ∈ ω < m (cid:1) is an s -indiscernible tree,(2) { ϕ i ( x, a η (cid:22) i ) : i < m } is consistent for every η ∈ ω m ,(3) (cid:8) ϕ i (cid:0) x, a η(cid:95) (cid:104) j (cid:105) (cid:1) : j ∈ ω (cid:9) is k i -inconsistent for every i < m − η ∈ ω i .For l < m and ν ∈ ω l we define ν ∗ = ( ν (0) , , ν (1) , , . . . , ν ( l − , ∈ ω < m .Let { ν , . . . , ν k − } ⊆ ω 1) and ν ∗ i (cid:48) (cid:22) ( l i (cid:48) − 1) are incomparable. Then by Lemma2.3(2) we see that the sequences ¯ a i = (cid:0) a ν ∗ i (cid:22) ( l i − (cid:98) (cid:104) j (cid:105) : j ∈ ω (cid:1) are mutually indis-cernible. But if (cid:8) ϕ l i (cid:0) x, a ν ∗ i (cid:1) : i < k (cid:9) was consistent, this would give us an inp-pattern of depth k , contrary to the assumption (as (cid:8) ϕ l i (cid:0) x, a ν ∗ i (cid:22) ( l i − (cid:98) (cid:104) j (cid:105) (cid:1) : j ∈ ω (cid:9) is k l i -inconsistent for every i ).Now using the claim it is easy to see that (cid:8) ϕ l ( η ) ( x, a η ∗ ) : η ∈ ω Let k < ω be fixed. Assume that for any n < ω we have, insome fixed number of variables, an (sct , k ) -pattern of depth n . Then there are, inthe same number of variables, (cdt , -patterns of arbitrary finite depth.Proof. Let m ∈ ω be arbitrary, and let ( a η : η ∈ ω Then ( b η : η ∈ ω Let κ ≤ ω , and let ( a η : η ∈ ω <κ ) , ( ϕ i ( x, y i ) : i < κ ) be a (cdt , -pattern (i.e. for every η ∈ ω <κ the set (cid:8) ϕ l ( η )+1 ( x, a η ˆ j ) : j ∈ ω (cid:9) is -inconsistent).For η ∈ ω <κ define b η = a η (cid:22) a η (cid:22) . . . a η (cid:22) ( l ( η ) − a η and ψ i ( x ; y i, , . . . , y i,i − ) = (cid:86) j
Proposition 3.10. If κ n cdt ≥ ℵ , then either κ n inp ≥ ℵ or κ n sct ≥ ℵ .Remark . Inspecting the proof, we actually get the following bound: κ n sct ≥ ( κ n cdt ) κn inp .The next proposition is an analog of Proposition 3.8 for inp-patterns. It is notused in this paper, but we include it for reference. Proposition 3.12. Let k < ω be fixed. Assume that for any n < ω we have, insome fixed number of free variables, an inp -pattern of depth n such that each rowis k -inconsistent. Then there are, in the same number of variables, inp -patterns ofarbitrary finite depths in which every row is -inconsistent.Proof. Let m ∈ ω be arbitrary, and let ( a i,j ) i Proposition 3.13. κ ncdt ≥ ℵ implies κ nsct ≥ ℵ .Proof. Suppose ( ϕ i : i < ω ), ( a η ) η ∈ ω <ω is a cdt-pattern. By replacing a η with b η = ( a ∅ , a η | , . . . , a η | l ( η ) − , a η ) and ϕ i ( x ; a η ) by ψ i ( x ; b η ) := (cid:94) j ≤ i ϕ j ( x ; a η | j ) , if necessary, we may assume that if ν (cid:67) η , then | = ( ∀ x )[ ϕ l ( η ) ( x ; a η ) → ϕ l ( ν ) ( x ; a ν )] . Then by replacing ( a η ) η ∈ ω <ω by an s -indiscernible tree locally based on it, we maymoreover assume the ( a η ) η ∈ ω <ω are s -indiscernible by Fact 2.5.By induction, we will construct cdt-patterns ( ϕ ni : i < ω ), ( a nη ) η ∈ ω <ω so that(1) ( a nη ) η ∈ ω <ω is s -indiscernible.(2) For all η ∈ ω Condition (5) follows by the inconsistency (3.14) and the definition of ψ n +1 . To see(6), we note that by the minimality of k , { ψ n +1+ j ( x ; b n (cid:95) (cid:104) i (cid:105) (cid:95) j ) : i < k − , j < ω } is consistent. By (3) above and the definition of the ψ m , this establishes (6).Let ( c η ) η ∈ ω <ω be the 2 k − -fold widening of ( b η ) η ∈ ω <ω at level n + 1. Let( χ i ( x ; w i ) : i < ω ) be defined as follows: if i < n + 1, let w i = z i and χ i ( x ; w i ) = ψ i ( x ; z i ). If i ≥ n + 1, let w i = ( z i , . . . , z k − − i ) a tuple of variables consisting of2 k − copies of z i . Then put χ i ( x ; w i ) = (cid:94) j< k − ψ i ( x ; z ji ) . By Lemma 7.3, ( c η ) η ∈ ω <ω is s -indiscernible and, by construction, ( χ i ( x ; w i ) : i < ω ),( c η ) η ∈ ω <ω is a cdt-pattern and, moreover, if i (cid:54) = j { χ n +1 ( x ; c n (cid:95) (cid:104) i (cid:105) ) , χ n +1 ( x ; c n (cid:95) (cid:104) j (cid:105) ) } is inconsistent. For all m < ω and η ∈ ω <ω , define ϕ n +1 m = ξ m and a n +1 η = c η .We have satisfied requirements (1)-(3) and since our construction did not modifythe formulas and parameters with level at most n , the construction never injuresrequirement (4).Finally, define a cdt-pattern ( ϕ ∞ n : n < ω ), ( a ∞ η ) η ∈ ω <ω by ϕ ∞ n = ϕ nn and a ∞ η = a l ( η ) η . Our construction gives(7) ( a ∞ η ) η ∈ ω <ω is s -indiscernible.(8) If η ∈ ω ω , { ϕ ∞ ( x ; a ∞ η | n ) : n < ω } is consistent.(9) If ν (cid:67) η , then | = ( ∀ x )[ ϕ ∞ l ( η ) ( x ; a ∞ η ) → ϕ ∞ l ( ν ) ( x ; a ∞ ν )] . (10) For all n , and i (cid:54) = j { ϕ ∞ n +1 ( x ; a ∞ n (cid:95) (cid:104) i (cid:105) ) , ϕ ∞ n +1 ( x ; a ∞ n (cid:95) (cid:104) j (cid:105) ) } is inconsistent.By s -indiscernibility, (9) and (10) imply that if η ⊥ ν , then { ϕ ∞ l ( η ) ( x ; a ∞ η ) , ϕ ∞ l ( ν ) ( x ; a ∞ ν ) } is inconsistent. This shows ( ϕ ∞ n : n < ω ) and ( a ∞ η ) η ∈ ω <ω form an sct -pattern. Wehave thus shown κ nsct ≥ ℵ . (cid:3) We obtain the main theorem of this section. Theorem 3.15. If T is countable, then κ cdt ( T ) = κ sct ( T ) + κ inp ( T ) . Moreover, κ n cdt ( T ) = κ n sct ( T ) + κ n inp ( T ) , provided κ n cdt ( T ) is infinite.Proof. By Observation 3.5, κ n cdt ( T ) ≥ n for any T and κ cdt ( T ) > | T | + if and only if κ cdt ( T ) = ∞ . It follows that, for countable theories, the possible values of κ cdt ( T ),and the only possible infinite values of κ n cdt ( T ), are ℵ , ℵ , and ∞ . The case of ℵ is treated in Proposition 3.10, ℵ is handled by Proposition 3.13, and for ∞ theresult follows from Shelah’s theorem (Fact 3.2). (cid:3) 4. TP and weak k − TP Say that a subset { η i : i < k } ⊆ ω <ω is a collection of distant siblings if given i (cid:54) = i (cid:48) , j (cid:54) = j (cid:48) , all of which are < k , η i ∧ η i (cid:48) = η j ∧ η j (cid:48) . Definition 4.1. Fix k ≥ ϕ ( x ; y ) has SOP if there is a collection of tuples ( a η ) η ∈ <ω satisfying the following. (a) For all η ∈ ω , { ϕ ( x ; a η | α ) : α < ω } is consistent.(b) If η, ν ∈ <ω and η ⊥ ν , then { ϕ ( x ; a η ) , ϕ ( x ; a ν ) } is inconsistent.(2) The formula ϕ ( x ; y ) has weak k-TP if there is a collection of tuples ( a η ) η ∈ ω <ω satisfying the following.(a) For all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent.(b) If { η i : i < k } ⊆ ω <ω is a collection of distinct distant siblings, then { ϕ ( x ; a η i ) : i < k } is inconsistent.(3) The formula ϕ ( x ; y ) has k-TP if there is a collection of tuples ( a η ) η ∈ ω <ω satisfying the following.(a) For all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent.(b) If { η i : i < k } ⊆ ω <ω is a collection of distinct pairwise incomparablenodes, then { ϕ ( x ; a η i ) : i < k } is inconsistent.(4) The theory T has either of the above properties if some formula does.We remark that TP is equivalent to SOP in a strong way: Fact 4.2. If a theory has TP witnessed by a formula ϕ , then the theory also hasSOP witnessed by the same formula, and vice versa.We recall the argument from [1]. Suppose ϕ ( x ; y ) witnesses SOP with respectto the tree of parameters ( b η ) η ∈ <ω . Define a map h : ω <ω → <ω recursively by h ( ∅ ) = ∅ and h ( β (cid:95) (cid:104) i (cid:105) ) = h ( β ) (cid:95) i (cid:95) 0, where 1 i denotes the all 1’s sequence oflength i . It is straightforward to check that ϕ ( x ; y ) witnesses TP with respect tothe parameters ( b h ( η ) ) η ∈ ω <ω . The converse is obvious. Although SOP and TP areequivalent, it will be important for us to notationally distinguish them, as variouscombinatorial constructions are simplified by a judicious choice of the index set.In [16], Kim and Kim show that k -TP is equivalent to TP for all k ≥ 2, butthe questions of whether weak k -TP is equivalent to TP was left unresolved.Using strongly indiscernible trees, we settle this, as well as show that TP is alwayswitnessed by a formula in a single free variable.4.1. Finding and manipulating indiscernible witnesses.Lemma 4.3. (1) If T has weak k-TP witnessed by ϕ ( x ; y ) then there is astrongly indiscernible tree ( a η ) η ∈ ω <ω witnessing this.(2) If ϕ ( x ; y ) has TP then there is a strongly indiscernible tree witnessing this.(3) If ϕ ( x, y ) has SOP , then there is a strongly indiscernible tree ( a η ) η ∈ <ω witnessing this.Proof. (1) This was observed in [23], but we sketch a proof here for completeness.Let ( b η ) η ∈ ω <ω be a tree of tuples with respect to which ϕ ( x ; y ) witnesses weak k -TP . Let ( a η ) η ∈ ω <ω be locally based on the tree ( b η ) η ∈ ω <ω . Suppose η , . . . , η n − ∈ ω <ω lie along a path and let ψ ( y , . . . , y n − ) denote the formula ( ∃ x ) (cid:86) i Lemma 4.6. (Path Collapse) Suppose κ is an infinite cardinal, ( a η ) η ∈ <κ is a treestrongly indiscernible over a set of parameters C and, moreover, ( a α : 0 < α < ω ) is indiscernible over cC . Let p ( y ; z ) = tp( c ; ( a (cid:95) γ : γ < κ ) /C ) . Then if p ( y ; ( a (cid:95) γ ) γ<κ ) ∪ p ( y ; ( a (cid:95) γ ) γ<κ ) is not consistent, then T has SOP , witnessed by a formula with free variables y .Proof. We may add C to the language, so assume C = ∅ . With p defined as above,suppose p ( y ; ( a (cid:95) γ : γ < κ )) ∪ p ( y ; ( a (cid:95) γ : γ < κ ))is inconsistent. Then by indiscernibility and compactness, there is a formula ψ and n < ω so that { ψ ( y ; a , . . . , a (cid:95) n − ) } ∪ { ψ ( y ; a , a , . . . , a (cid:95) n − ) } is inconsistent. Let ( b η ) η ∈ <κ denote the n -fold elongation of ( a η ) η ∈ <κ . By Lemma2.7, ( b η : η ∈ <κ ) is strongly indiscernible. Since c | = { ψ ( y ; b α ) : α < κ } and ψ ( y ; b ) ∧ ψ ( y ; b ) is inconsistent (by strong indiscernibility), by Lemma 4.5, ψ witnesses SOP . (cid:3) Remark . It is significant that the type p does not contain a ∅ as a parameter.As b and b are incomparable and ψ ( x ; b ) and ψ ( x ; b ) are inconsistent, we canconclude that ψ ( x ; b η ) and ψ ( x ; b ν ) are inconsistent for all incomparable η, ν bystrong indiscernibility. But, for example, strong indiscernibility does not guarantee b (cid:95) b (cid:95) has the same type as b b over a ∅ as 0 ∧ ∅ while 0 n − (cid:95) ∧ n − (cid:95) n − .We now give two applications of the path-collapse lemma.4.2. Weak k − TP .Theorem 4.8. Given k ≥ , T has weak k -TP if and only if T has TP .Proof. We will show that if T has weak k -TP , then T has SOP . Let ϕ ( x ; y )witness weak k -TP with respect to the strongly indiscernible tree ( a η ) η ∈ ω <ω . Let n be maximal so that { ϕ ( x ; a (cid:104) i (cid:105) (cid:95) α ) : i < n, α < ω } is consistent. By definition of weak k -TP , n is at least 1 and at most k − 1. Let C = { a (cid:104) i (cid:105) (cid:95) α : i < n − , α < ω } (and put C = ∅ in the case that n = 1). Given η ∈ ω <ω , let ˆ η be defined byˆ η ( i ) = (cid:26) η ( i ) + n − i = 0 η ( i ) otherwise , for all i < l ( η ). The tree ( b η ) η ∈ ω <ω defined by b η = a ˆ η is strongly indiscernible over C . By choice of n , { ϕ ( x ; a (cid:104) i (cid:105) (cid:95) α ) : i < n, α < ω } is consistent, so let c realize it. By compactness, Ramsey, and automorphism, wemay assume ( b α : 0 < α < ω ) (i.e. ( a (cid:104) n − (cid:105) (cid:95) α : α < ω )) is indiscernible over c .Letting the type p be defined by p ( y ; z ) = tp( c ; ( b (cid:95) α : α < α ) /C ) , and unravelling definitions, we see that the type p ( y ; ( b (cid:95) α : α < ω )) ∪ p ( y ; ( b (cid:95) α : α < ω ))implies { ϕ ( x ; a (cid:104) i (cid:105) (cid:95) α ) : i < n + 1 , α < ω } and is therefore inconsistent by the choiceof n . By path-collapse, we’ve shown that T has SOP , completing one direction.The other direction is obvious. (cid:3) Reducing to one variable.Proposition 4.9. Suppose T witnesses SOP via ϕ ( x, y ; z ) . Then there is a for-mula ϕ ( x ; v ) with free variables x and parameter variables v , or a formula ϕ ( y ; w ) with free variables y and parameter variables w so that one of ϕ and ϕ witness SOP .Proof. Let ϕ ( x, y ; z ) witness SOP with respect to the strongly indiscernible tree( a η ) η ∈ <ω . The first path is consistent and it is an indiscernible sequence so itfollows that there is some ( c, c ) | = { ϕ ( x, y ; a α ) : α < ω } and such that moreover( a α : α < ω ) is indiscernible over c (by Ramsey, automorphism, and compactness). N MODEL-THEORETIC TREE PROPERTIES 15 Define the function h : 2 <ω → <ω recursively by h ( ∅ ) = ∅ and h ( η (cid:95) (cid:104) i (cid:105) ) = h ( η ) (cid:95) (cid:95) (cid:104) i (cid:105) . Define the tree ( b η ) η ∈ <ω by b η = a h ( η ) . It is proved in Lemma7.7(1) that ( b η ) η ∈ <ω is a strongly indiscernible tree. For each n , define a map h n : 2 <ω → <ω by h n ( η ) = (cid:26) h ( η ) if l ( η ) ≤ nh ( ν ) (cid:95) ξ if η = ν (cid:95) ξ, l ( ν ) = n. By Lemma 7.7(2), the tree ( d n,η ) η ∈ <ω defined by d n,η = a h n ( η ) is strongly indis-cernible as well. Moreover, as paths in ( b η ) η ∈ <ω and ( d n,η ) η ∈ <ω are containedin paths in ( a η ) η ∈ <ω and incomparable elements in these trees correspond to in-comparable elements in ( a η ) η ∈ <ω , ϕ witnesses SOP with respect to these trees ofparameters as well.Assume that no formula in the variable y has SOP . By induction, we will choose c n so that(*) { ϕ ( x, c n ; d n,η | m ) : m < n } ∪ { ϕ ( x, c n ; d n,η(cid:95) α ) : α < ω } is consistent for every η ∈ ≤ n .For this, consider ( d ( n ) n,η ) η ∈ <ω , the n th-fattening of ( d n,η ), and let C n = ( d n,η : η ∈ There is c n +1 such that (cid:16) ( d ( n +1) n +1 , α ) : α < ω (cid:17) is indiscernible over c n +1 C n and c n (cid:16) d ( n ) n, (cid:95) (cid:95) α (cid:17) ≡ d ( n ) n, ∅ C n c n +1 (cid:16) d ( n ) n, (cid:95) (cid:95) α (cid:17) ≡ d ( n ) n, ∅ C n c n +1 (cid:16) d ( n ) n, (cid:95) (cid:95) α (cid:17) . Note that d ( n ) n, ∅ C n = C n +1 . Proof: The base case is above. Let p n ( y, z ) = tp (cid:16) c n , ( d ( n ) n, (cid:95) (cid:95) α : α < ω ) / ( d n, ∅ ) ( n ) C n (cid:17) . By the path-collapse lemma, p n (cid:16) y, (cid:16) ( d ( n ) n, (cid:95) (cid:95) α ) : α < ω (cid:17)(cid:17) ∪ p n (cid:16) y, (cid:16) ( d ( n ) n, (cid:95) (cid:95) α ) : α < ω (cid:17)(cid:17) is consistent. Let c n +1 realize it. Moreover, as (cid:16) d ( n ) n, (cid:95) (cid:95) α , d ( n ) n, (cid:95) (cid:95) α (cid:17) α<ω = (cid:16) d ( n +1) n +1 , α (cid:17) α<ω is an indiscernible sequence, by Ramsey, automorphism, and compactness we mayassume that it is indiscernible over c n +1 C n . This shows (*).By the definition of the trees ( d n,η ) η ∈ <ω , we have shown that { ϕ ( x, c n ; b η | m ) : m < n } ∪ { ϕ ( x, c n ; b η(cid:95) α ) : α < ω } is consistent for each n and η ∈ ≤ n . By compactness, we can find one c which worksfor all possible paths in 2 ω simultaneously, giving a tree ( c, b η ) η ∈ <ω witnessingSOP for ϕ ( x ; y, z ). (cid:3) Remark . The necessity of defining the trees ( b η ) η ∈ <ω and ( d n,η ) η ∈ <ω via h and h n , respectively, stems from a technical obstacle in applying the path-collapselemma: starting with the tree ( a η ) η ∈ <ω , we cannot apply the path collapse lemmadirectly to the type q ( y ; ( a α : α < ω )) = tp( c / ( a α : α < ω )) , as this type has a ∅ as a parameter (see Remark 4.7 above). This is corrected bythe offset functions h and h n , allowing us to apply the path-collapse lemma ‘higher’in the tree, where the parameters of interest are indiscernible over what we haveconstructed so far. Corollary 4.11. (1) T has SOP if and only if there is some formula in asingle free variable witnessing this(2) T has TP if and only if there is some formula in a single free variablewitnessing this At this point it is natural to ask if κ = κ n sct holds for arbitrary n , at least forcountable theories. Corollary 4.11 resolves the case of ∞ , and we remark that thecase of ℵ follows from a standard argument in simplicity theory. Proposition 4.12. Any theory satisfies κ = κ n cdt , for all n ∈ ω .Proof. The following are equivalent (see e.g. [4, Proposition 3.8]).(1) κ n cdt ≤ κ .(2) For any type p ( x ) ∈ S n ( A ), there is some A ⊆ A such that | A | < κ and p does not divide over A .Clearly κ n cdt ≥ κ . Assume now that κ ≤ κ for some κ . We show by inductionthat (2) above holds for all n with respect to κ . Given a . . . a n a n +1 and A , itfollows by the inductive assumption that a . . . a n | (cid:94) A A for some A ⊆ A with | A | < κ and a n +1 | (cid:94) A a ...a n Aa . . . a n for some A ⊆ A with | A | < κ . Combinedthis implies (by left transitivity and right base monotonicity of dividing in arbitrarytheories, see e.g. [9, Section 2]) that a . . . a n a n +1 | (cid:94) A A A and | A ∪ A | < κ . (cid:3) Corollary 4.13. If κ n sct ≥ ℵ then κ ≥ ℵ .Proof. By Proposition 3.13, it is enough to show that κ ≥ ℵ , which follows byassumption and Proposition 4.12. (cid:3) The case of ℵ appears to involve more complicated combinatorics and we leaveit for future work.5. Independence and amalgamation in NSOP theories We recall the definition of SOP from [22]: Definition 5.1. A formula ϕ ( x ; y ) exemplifies SOP if and only if there are ( a η ) η ∈ <ω so that • For all η ∈ ω , { ϕ ( x ; a η | n ) : n < ω } is consistent, • If η (cid:95) (cid:69) ν ∈ <ω , then { ϕ ( x ; a η(cid:95) ) , ϕ ( x ; a ν ) } is inconsistent.Given an array ( c i,j ) i<ω,j< , write c i = ( c i, , c i, ) and c
Suppose ( c i,j ) i<ω,j< is an array and ϕ ( x ; y ) is a formula over C with(1) For all i < ω , c i, ≡ Cc
Proof. For each n , define a subtree T n of 2 <ω by T n = { η (cid:95) α : η ∈ ≤ n , α < ω } ∪ { η (cid:95) α (cid:95) η ∈ ≤ n , α < ω } . Let P ( T n ) ⊆ ω be the set of infinite branches of T n . Namely, P ( T n ) = { η (cid:95) ω : η ∈ ≤ n } . As a first step, by induction on n we build an ascending sequence of trees ( l η , r η ) η ∈ T n ,so that:(1) if η ∈ P ( T n ), ( l η | α , r η | α ) α<ω ≡ C ( c α, , c α, ) α<ω ,(2) if η (cid:95) ∈ T n then r η(cid:95) = l η(cid:95) ,(3) if η ∈ ≤ n then ( l η(cid:95) , r η(cid:95) ) ≡ Cl (cid:69) η r (cid:69) η ( l η(cid:95) , r η(cid:95) ).For the n = 0 case, define l α = c α, , r α = c α, and l α (cid:95) = r α (cid:95) for all α < ω .For each α < ω , we can choose σ α ∈ Aut( M /Cc <α ) such that σ α ( c α, ) = c α, . Let r α (cid:95) = σ α +1 ( c α +1 , ) = σ α +1 ( r α (cid:95) ). This defines ( l η , r η ) η ∈ T satisfying (1)-(3).Now by induction suppose ( l η , r η ) η ∈ T n has been defined. Suppose η ∈ P ( T n +1 ) \ P ( T n ). Then there is ν ∈ ≤ n so that η = ν (cid:95) (cid:95) ω . Then ν (cid:95) ∈ T n and, byinduction, ( l ν(cid:95) , r ν(cid:95) ) ≡ Cl (cid:69) ν r (cid:69) ν ( l ν(cid:95) , r ν(cid:95) )and r ν(cid:95) = l ν(cid:95) . Choose an automorphism σ ∈ Aut( M /Cl (cid:69) ν r (cid:69) ν ) such that σ ( l ν(cid:95) , r ν(cid:95) ) = l ν(cid:95) , r ν(cid:95) . Then define( l ν(cid:95) (cid:95) α , r ν(cid:95) (cid:95) α ) = σ ( l ν(cid:95) (cid:95) α , r ν(cid:95) (cid:95) α ) and( l ν(cid:95) (cid:95) α (cid:95) , r ν(cid:95) (cid:95) α (cid:95) ) = σ ( l ν(cid:95) (cid:95) α (cid:95) , r ν(cid:95) (cid:95) α (cid:95) )for all α < ω . This completes the construction of ( l η , r η ) η ∈ T n +1 , properties (1)–(3)are satisfied because of the inductive assumption. We obtain ( l η , r η ) η ∈ <ω as theunion over all n of ( l η , r η ) η ∈ T n .Now we check that with respect to the parameters ( l η ) η ∈ <ω , ϕ witnesses SOP .Fix any path η ∈ ω , we have to check that { ϕ ( x ; l η | α ) : α < ω } is consistent.But given any n , l (cid:69) ( η | n ) ⊂ T n and by (1), l (cid:69) ( η | n ) ≡ C ( c α, ) α ≤ n hence { ϕ ( x ; l η | α ) : α ≤ n } is consistent, as { ϕ ( x ; c α, ) : α ≤ n } is consistent, by hypothesis. Then { ϕ ( x ; l η | α ) : α < ω } is consistent by compactness.Now fix η ⊥ ν ∈ <ω so that ( η ∧ ν ) (cid:95) (cid:69) η and ( η ∧ ν ) (cid:95) ν . Wemust check { ϕ ( x ; l η ) , ψ ( x ; l ν ) } is inconsistent. As ν = ( η ∧ ν ) (cid:95) 1, we know that l ν = l ( η ∧ ν ) (cid:95) = r ( η ∧ ν ) (cid:95) by (2). Let ξ = ( η ∧ ν ) (cid:95) 0. Then ξ (cid:69) η and l ν = r ξ soit suffices to show { ϕ ( x ; l η ) , ϕ ( x ; r ξ ) } is inconsistent. Let n = l ( η ) and m = l ( ξ ).Then m ≤ n and by (1), we have ( l η , r ξ ) ≡ C ( c n, , c m, ). By hypothesis, this implies { ϕ ( x ; l η ) , ϕ ( x ; r ξ ) } is inconsistent, so we finish. (cid:3) Definition 5.3. Suppose | (cid:94) is an Aut( M )-invariant ternary relation on small sub-sets of M .(1) We say | (cid:94) satisfies weak independent amalgamation over models if, given M | = T , b c ≡ M b c satisfying b i | (cid:94) M c i for i = 0 , c | (cid:94) M c , thereis b satisfying bc ≡ M bc ≡ M b c .(2) We say | (cid:94) satisfies independent amalgamation over models if, given M | = T , b ≡ M b satisfying b i | (cid:94) M c i for i = 0 , c | (cid:94) M c , there is b satisfying bc ≡ M b c and bc ≡ M b c .(3) We say | (cid:94) satisfies stationarity over models if: given M | = T , if b ≡ M b and b | (cid:94) M c, b | (cid:94) M c then b ≡ Mc b . Definition 5.4. Suppose A, B, C are small subsets of the monster M .(1) We say A | (cid:94) iC B if and only if tp( A/BC ) can be extended to a global typeLascar-invariant over C . We denote its dual by | (cid:94) ci - i.e. A | (cid:94) iC B holds ifand only if B | (cid:94) ciC A .(2) We say A | (cid:94) uC B if and only if tp( A/BC ) is finitely satisfiable in C . Wedenote its dual by | (cid:94) h - i.e. A | (cid:94) hC B if and only if B | (cid:94) uC A .Suppose q ( x ) and r ( y ) are global M -invariant types. Recall that the product q ( x ) ⊗ r ( y ) ∈ S xy ( M ) is defined by q ( x ) ⊗ r ( y ) = tp( ab/ M ) where b | = r and a | = q | M b . Proposition 5.5. Fix a model M | = T . Suppose c | (cid:94) iM c , c j | (cid:94) iM b j for j = 0 , and b c ≡ M b c , but there is no b such that bc ≡ M bc ≡ M b c . Then T has SOP .Proof. Let p ( x ; y ) = tp( b c /M ). Our assumption entails that p ( x ; c ) ∪ p ( x ; c ) isinconsistent. By compactness, there is some ϕ ( x ; y ) ∈ p ( x ; y ) so that { ϕ ( x ; c ) , ϕ ( x ; c ) } is inconsistent. Fix a global M -invariant type r so that c | = r | M b and a global M -invariant type q so that c | = q | M c . Then c c | = ( q ⊗ r ) | M . Let ( c i , c i ) ≤ i<ω be a Morley sequence in ( q ⊗ r ) | Mb c c and put ( c , c ) = ( c , c ).First, we note that b | = { ϕ ( x ; c i ) : i < ω } so a fortiori { ϕ ( x ; c i ) : i < ω } isconsistent. Secondly, for any N < ω , we have( c c ) . . . ( c N c N ) i | (cid:94) M c c so by M -invariance and the fact that c ≡ M c , we know that c ≡ Mc c ...c N c N c Next, as c | = q | Mc c , we have c ≡ Mc c and therefore { ϕ ( x ; c ) , ϕ ( x ; c ) } isinconsistent. As ( c i , c i ) i<ω is an M -indiscernible sequence, we’ve shown the fol-lowing.(1) If X ⊆ ω and j < k for all k ∈ X , then { ϕ ( x ; c k ) : k ∈ X } ∪ { ϕ ( x ; c ji ) } isconsistent for i = 0 , X ⊆ ω and j < k for all k ∈ X , then, writing c X for an enumeration of { c k c k : k ∈ X } , we have c j ≡ Mc X c j .(3) If j ≤ k then { ϕ ( x ; c j ) , ϕ ( x ; c k ) } is inconsistent.Now by compactness (reversing the ordering on the sequence of pairs), we can findan array ( d i,j ) i<ω,j< such that the following holds.(1) For all i < ω , d i, ≡ Md
Proposition 5.6. Assume ϕ ( x ; y ) witnesses SOP . Then there are M , c , c , b , b so that c | (cid:94) uM c , c | (cid:94) uM b , c | (cid:94) uM b , b c ≡ M b c and | = ϕ ( b , c ) ∧ ϕ ( b , c ) but ϕ ( x ; c ) ∧ ϕ ( x ; c ) is inconsistent. N MODEL-THEORETIC TREE PROPERTIES 19 Proof. Suppose T has SOP witnessed by ϕ . By compactness, we may assume thatwe have a tree of tuples ( a η ) η ∈ <κ for κ large enough ( ≥ | T | suffices) so that • For all η ∈ κ , { ϕ ( x ; a η | α ) : α < κ } is consistent • η (cid:95) (cid:67) ν ∈ <κ , then { ϕ ( x ; a η(cid:95) ) , ϕ ( x ; a ν ) } is inconsistent.Fix a Skolemization T Sk of T and in what follows, we’ll work modulo this expandedtheory. We will construct a sequence ( η i , ν i ) i<ω of elements of 2 <κ satisfying thefollowing.(1) a ν i , a η i have the same type over a η
1. Now (1) and (2) are clearly satisfied, and, as α < β , ( η n ∧ ν n ) = η n − (cid:95) α so (3) follows. This completes the construction.Now we claim that ( a η i , a ν i ) i<ω satisfies:(4) { ϕ ( x ; a η i ) : i < ω } is consistent,(5) a ν i , a η i have the same type over a ν
The following are equivalent.(1) | (cid:94) ci satisfies weak independent amalgamation: given any M | = T , b c ≡ M b c so that c | (cid:94) iM c and c j | (cid:94) iM b j for j = 0 , , there is b so that bc ≡ M bc ≡ M b c .(2) | (cid:94) h satisfies weak independent amalgamation: given any M | = T , b c ≡ M b c so that c | (cid:94) uM c and c j | (cid:94) uM b j for j = 0 , , there is b so that bc ≡ M bc ≡ M b c .(3) T is NSOP .Proof. (1) = ⇒ (2) is clear.(2) = ⇒ (3) is Proposition 5.6.(3) = ⇒ (1) is Proposition 5.5. (cid:3) Proposition 5.8. Assume there is an Aut ( M ) -invariant independence relation | (cid:94) on small subsets of the monster M | = T such that it satisfies the following properties,for an arbitrary M | = T and arbitrary tuples from M . (1) Strong finite character: if a (cid:54) | (cid:94) M b , then there is a formula ϕ ( x, b, m ) ∈ tp ( a/bM ) such that for any a (cid:48) | = ϕ ( x, b, m ) , a (cid:48) (cid:54) | (cid:94) M b .(2) Existence over models: M | = T implies a | (cid:94) M M for any a .(3) Monotonicity: aa (cid:48) | (cid:94) M bb (cid:48) = ⇒ a | (cid:94) M b .(4) Symmetry: a | (cid:94) M b ⇐⇒ b | (cid:94) M a .(5) Independent amalgamation: c | (cid:94) M c , b | (cid:94) M c , b | (cid:94) M c , b ≡ M b im-plies there exists b with b ≡ c M b , b ≡ c M b .Then T is NSOP .Proof. Claim Let M | = T , then a | (cid:94) uM b = ⇒ a | (cid:94) M b . Proof of claim. If a (cid:54) | (cid:94) M b then by strong finite character, there is some ϕ ( x ; m, b ) ∈ tp( a/M b ) so that a (cid:48) (cid:54) | (cid:94) M b for any a (cid:48) with | = ϕ ( a (cid:48) ; m, b ). However, as a | (cid:94) uM b , it fol-lows that there is some a (cid:48) ∈ M such that | = ϕ ( a (cid:48) ; m, b ). Then b (cid:54) | (cid:94) M a (cid:48) by symmetryand b (cid:54) | (cid:94) M M by monotonicity, contradicting existence.Now assume towards contradiction that T has SOP , and let M, c , c , b , b , ϕ ( x ; y )as given in Proposition 5.6. By the claim and symmetry of | (cid:94) we have c | (cid:94) M c , b | (cid:94) M c , b | (cid:94) M c . As | (cid:94) satisfies independent amalgamation over models, thereis some b | (cid:94) M c c , b ≡ c M b , b ≡ c M b . This contradicts the inconsistency of { ϕ ( x ; c ) , ϕ ( x ; c ) } . Remark . (1) We don’t require the local character here, as it would thengive simplicity according to the theorem of Kim and Pillay [18].(2) We do require strong finite character, which is not required in Adler’s def-inition of mock stability and mock simplicity (see [2, the discussion afterDefinition 12]). Indeed, there are mock stable examples arbitrarily high inthe SOP n hierarchy.6. Examples of NSOP theories Vector spaces with a generic bilinear form. Let L denote the languagewith two sorts V and K containing the language of abelian groups for variablesfrom V , the language of rings for variables from K , a function · : K × V → V , anda function [ ] : V × V → K . T ∞ is the model companion of the L -theory assertingthat K is a field, V is a K -vector space of infinite dimension with the action of K given by · , and [ ] is a non-degenerate bilinear form on V . If ( K, V ) | = T ∞ then K is an algebraically closed field.The theory T ∞ was introduced by Nicolas Granger in [12], who observed that itscompletions are not simple, but nonetheless have a notion of independence called Γ-non-forking satisfying essentially all properties of forking in stable theories, exceptlocal character. Definition 6.1. We are using the notation from [12, Notation 9.2.4]. Let M =( V, ˜ K ) be a sufficiently saturated model of T ∞ . Let A ⊆ B ⊂ M and c ∈ M with c a singleton. Let c | (cid:94) Γ A B be the assertion that K Ac | (cid:94) K A K B in the senseof non-forking independence for algebraically closed fields and one of the followingholds:(1) c ∈ ˜ K (2) c ∈ (cid:104) A (cid:105) (3) c (cid:54)∈ (cid:104) B (cid:105) and [ c, B ] is Φ-independent over A , N MODEL-THEORETIC TREE PROPERTIES 21 where ‘[ c, B ] is Φ-independent over A’ means that whenever { b , . . . , b n − } is alinearly independent set in B V ∩ ( V \ (cid:104) A (cid:105) ) then the set { [ c, b ] , . . . , [ c, b n − ] } isalgebraically independent over the field K B ( K Ac ).By induction, for c = ( c , . . . , c m ) define c | (cid:94) Γ A B by c Γ | (cid:94) A B ⇐⇒ ( c , . . . , c m − ) Γ | (cid:94) A B and c m Γ | (cid:94) Ac ...c m − Bc . . . c m − . Fact 6.2. [12, Theorem 12.2.2] Let M = ( V, K ) | = T ∞ . Then the relation onsubsets of M given by Γ-non-forking is automorphism invariant, symmetric, andtransitive. Moreover, it satisfies extension, finite character, and stationarity over amodel. Lemma 6.3. If c is a tuple and A, B are small sets with c (cid:54) | (cid:94) Γ A B , then there is aformula ϕ ( x ; a, b ) ∈ tp ( c/AB ) so that | = ϕ ( c (cid:48) ; a, b ) = ⇒ c (cid:48) Γ (cid:54) | (cid:94) A B. Proof. Suppose c = ( c , . . . , c n − ) a tuple and c (cid:54) | (cid:94) Γ A B . Let k be maximal sothat ( c , . . . , c k − ) | (cid:94) Γ A B . It follows that c k (cid:54) | (cid:94) Γ Ac ...c k − Bc . . . c k − , so one of thefollowing possibilities occurs:(1) K Ac ...c k (cid:54) | (cid:94) ACF K Ac ...ck − K Bc ...c k − (2) c k ∈ (cid:104) Bc . . . c k − (cid:105) \ (cid:104) Ac . . . c k − (cid:105) (3) There is a linearly independent set { d , . . . , d l − } from ( Bc . . . c k − ) V ∩ ( V \ (cid:104) Ac . . . c k − (cid:105) ) so that { [ c k , d ] , . . . , [ c k , d l − ] } is not algebraically in-dependent over K Bc ...c k − ( K Ac ...c k ).The existence of the desired formula requires an argument only in case (3). In thiscase, there is a nonzero polynomial p ( x , . . . , x l − ; a, b, c , . . . , c k ) with coefficientsin K Bc ...c k − ( K Ac ...c k ) so that p ([ c k , d ] , . . . , [ c k , d l − ]; a, b, c , . . . , c k ) = 0. Byreindexing the d j , we may assume that there is m ≤ l so that d j = c i j for j < m and d j ∈ B for j ≥ m . Let d = ( d m , . . . , d l − ). Writing y = ( y , . . . , y k ), let χ ( y ; a, b, d ) be the formula which asserts the following:(1) the polynomial p ( x , . . . , x l − ; a, b, y ) is a nonzero polynomial;(2) the set { y i , . . . , y i m − } ∪ { d m , . . . , d l − } is linearly independent;(3) p ([ y k , y i ] , . . . , [ y k , y i m − ] , [ y k , d m ] , . . . , [ y k , d l − ]; a, b, y ) = 0Then χ ( y ; a, b, d ) ∈ tp( c/B ) and if | = χ ( c (cid:48) ; a, b, d ) then it is easy to check c (cid:48) (cid:54) | (cid:94) Γ A B . (cid:3) Corollary 6.4. The two-sorted theory T ∞ of infinite dimensional vector spacesover algebraically closed fields with a generic bilinear form is NSOP . ω -free PAC fields of characteristic zero.Definition 6.5. A field F is called pseudo-algebraically closed if every absolutelyirreducible variety defined over F has an F -rational point. A field F is called ω -free if it has a countable elementary substructure F with G ( F ) ∼ = ˆ F ω , the free profinitegroup on countably many generators. In [5], Chatzidakis showed that a PAC field has a simple theory if and only if ithas finitely many degree n extensions for all n so an ω -free PAC field will not besimple. Nonetheless, she showed that an ω -free PAC field comes equipped with anotion of independence which is well-behaved. Fact 6.6. [6, 7] Suppose F is a sufficiently saturated ω -free PAC field of charac-teristic zero. Given A = acl( A ), B = acl( B ), C = acl( C ) with C ⊆ A, B ⊆ F ,write A | (cid:94) IC B to indicate that A | (cid:94) ACF C B and A alg B alg ∩ acl( AB ) = AB . Ex-tend this to non-algebraically closed sets by stipulating a | (cid:94) ID b holds if and only ifacl( aD ) | (cid:94) I acl( D ) acl( bD ). Then | (cid:94) I satisfies existence over models, monotonicity,symmetry, and independent amalgamation over models.It remains to check that | (cid:94) I satisfies strong finite character. The proof of it waspointed out to us by Zo´e Chatzidakis, whom we would like to thank. Lemma 6.7. Suppose F is a sufficiently saturated ω -free PAC field of characteristiczero. If a, b, c are tuples from F and a | (cid:94) Ic b then there is a formula ϕ ( x ; b, c ) ∈ tp ( a/bc ) so that if F | = ϕ ( a (cid:48) ; b, c ) then a (cid:48) (cid:54) | (cid:94) Ic b .Proof. If a (cid:54) | (cid:94) ACF c b , then the existence of such a formula is clear, so we may as-sume a | (cid:94) ACF c b . As a (cid:54) | (cid:94) Ic b , there are β ∈ (cid:104) cb (cid:105) alg , α ∈ (cid:104) ca (cid:105) alg not in F such that F ( α ) = F ( β ) and β / ∈ F (cid:104) c (cid:105) alg . We choose them so that F ( β ) is Galois over F (always possible since F ∩ (cid:104) ca (cid:105) alg (cid:104) cb (cid:105) alg is Galois over ( F ∩ (cid:104) ca (cid:105) alg )( F ∩ (cid:104) cb (cid:105) alg ) =acl( ca ) acl( cb )).Some of the conjugates of β over (cid:104) cb (cid:105) might lie in F (cid:104) c (cid:105) alg and this will be wit-nessed by elements of acl( cb ) = F ∩ (cid:104) cb (cid:105) alg . We choose an element b (cid:48) of acl( cb ) suchthat (cid:104) cbb (cid:48) (cid:105) contains (cid:104) cbβ (cid:105) ∩ F and (cid:104) cbb (cid:48) (cid:105) is closed under Aut(acl( cb ) / (cid:104) cb (cid:105) ). Let theformula θ ( y ; b, c ) isolate tp( b (cid:48) /bc ).Let P ( Y, b, c ) be a minimal polynomial of b (cid:48) over (cid:104) bc (cid:105) , and let Q ( Z, Y, b, c ) besuch that Q ( Z, b (cid:48) , b, c ) is a minimal polynomial of β over (cid:104) cbb (cid:48) (cid:105) . Claim. If | = θ ( b , b, c ), then P ( b , b, c ) = 0, Q ( Z, b , b, c ) is irreducible of degree[ (cid:104) cbβ (cid:105) : (cid:104) cbb (cid:48) (cid:105) ] and a solution of Q defines a Galois extension, which is not containedin F (cid:104) c (cid:105) alg .The first two assertions of the claim are immediate. For the last one, assumethat ( b , b ) satisfies P ( b , b, c ) = 0 ∧ Q ( b , b , b, c ) = 0, and that Q ( Z, b , b, c ) isirreducible and defines a Galois extension of the right degree (all this is expressiblein tp F ( b (cid:48) /bc )), but that b ∈ F (cid:104) c (cid:105) alg . Then there is a formula in tp F ( b /cb ) whichwill say that such a b exists, and is therefore not in tp F ( b (cid:48) /bc ).Similarly let a (cid:48) ∈ acl( ac ) be such that (cid:104) caα (cid:105) ∩ F = (cid:104) caa (cid:48) (cid:105) and let R ( W, T, c ) besuch that R ( W, a, c ) is a minimal polynomial of a (cid:48) over (cid:104) ca (cid:105) and let S ( X, W, T, c )be such that S ( X, a (cid:48) , a, c ) is a minimal polynomial of α over (cid:104) caa (cid:48) (cid:105) .The formula ϕ ( t, b, c ) is a conjunction of the following assertions: • ∃ yθ ( y, b, c ), • R ( W, t, c ) is not the trivial polynomial, • ( ∃ w ) R ( w, t, c ) = 0 and S ( X, w, t, c ) is irreducible over F of degree [ (cid:104) caα (cid:105) : (cid:104) caa (cid:48) (cid:105) ], • ( ∀ z )[ Q ( z, y, b, c ) = 0 → “ F ( z ) contains a root of S ( X, w, t, c ) = 0”. N MODEL-THEORETIC TREE PROPERTIES 23 These statements are first-order using standard facts on interpretability of finitealgebraic extensions of a field in a field and definability of irreducibility (see e.g.[24]).Assume now that d satisfies ϕ ( t, b, c ). Let y = b and w = d ∈ F be asguaranteed to exist by ϕ , and let b be a root of Q ( Z, b , b, c ) = 0; then F ( b ) isa proper Galois extension of F of degree [ (cid:104) cbβ (cid:105) : (cid:104) cbb (cid:48) (cid:105) ] which is not contained in F (cid:104) c (cid:105) alg .Because d satisfies ϕ , if d satisfies S ( X, d , d, c ) = 0, then F ( d ) = F ( b ). As F ( b ) (cid:54)⊆ F (cid:104) c (cid:105) alg , we necessarily have d (cid:54)∈ (cid:104) c (cid:105) alg and, therefore, either d (cid:54) | (cid:94) ACF c b or,otherwise, (cid:104) cd (cid:105) alg (cid:104) cb (cid:105) alg ∩ F (cid:54) = acl( cd )acl( cb ). This shows d (cid:54) | (cid:94) Ic b . (cid:3) Corollary 6.8. The theory of ω -free PAC fields of characteristic 0 is NSOP . Examples via Parametrization. In this subsection, we show how to con-struct NSOP theories from simple ones. We start with a simple theory T obtainedas the theory of a Fra¨ıss´e limit satisfying the strong amalgamation property and,by analogy with the theory of parametrized equivalence relations T ∗ feq , form theparametrization of this structure. We show that the resulting theories are NSOP by proving an independence theorem for a natural independence notion associatedto these theories. The construction we perform here was studied by Baudisch [3]in the context of arbitrary model complete theories eliminating ∃ ∞ . We expectthat our results hold in this greater generality as well, but our setting alreadyencompasses many interesting examples and simplifies the study of amalgamation.We begin by recalling some facts from Fra¨ıss´e theory. Definition 6.9. (SAP) Suppose K is a class of finite structures. We say K hasthe Strong Amalgamation Property (SAP) if given A, B, C ∈ K and embeddings e : A → B and f : A → C there is a D ∈ K and embeddings g : B → D and h : C → D so that the following diagram commutes: B g (cid:32) (cid:32) A e (cid:63) (cid:63) f (cid:31) (cid:31) DC h (cid:62) (cid:62) and, moreover, (im g ) ∩ (im h ) = im ge (and hence = im hf , as well).The following is a useful criterion for SAP: Fact 6.10. [14] Suppose K is the age of a countable ultrahomogeneous structure M . Then the following are equivalent:(1) K has the strong amalgamation property.(2) M has no algebraicity.Let K denote a Fra¨ıss´e class in a finite relational language L = (cid:104) R i : i < k (cid:105) whereeach relation symbol R i has arity n i . Let T the complete L -theory of the Fra¨ıss´elimit of K . We’ll define a new language L pfc which contains two sorts P and O .For each i < k , there is an ( n i + 1)-ary relation symbol R ix where x is a variable ofsort P and the suppressed n i variables belong to the sort O . Given an L pfc -structure M , it is convenient to write M = ( A, B ) where O ( M ) = A and P ( M ) = B . We will refer to elements named by O as objects and elementsnamed by P as parameters . Given b ∈ B , we define the L-structure associated to bin M , denoted A b , to be the L -structure interpreted in M with domain A and eachrelation symbol R i interpreted by R ib ( A ). If b ∈ B and C ⊆ A , write (cid:104) C (cid:105) b to denotethe L -substructure of A b generated by C (as we assume the language is relational,this will have C as its domain).We describe a class of finite structures K pfc to be the class defined in the followingway. Let K pfc = { M = ( A, B ) ∈ Mod( L pfc ) : | M | < ℵ , ( ∀ b ∈ B )( ∃ D ∈ K ) ( A b ∼ = D ) } From now on, we’ll assume K also satisfies SAP. Lemma 6.11. K pfc is a Fra¨ıss´e class satisfying the Strong Amalgamation Property(SAP).Proof. HP is clear and, as we allow the empty structure to be a model in K pfc , JEPfollows from SAP. So we show SAP.First, we may assume that 3 models in the amalgamation diagram have thesame set of parameters. Suppose ( A, D ), ( B, E ) and ( C, F ) are in K pfc and wehave embeddings ( C, F )( A, D ) i (cid:58) (cid:58) j (cid:36) (cid:36) ( B, E )By moving F and E over D if necessary, we may assume that i and j are just theinclusion maps on parameters and that F ∩ E = D . By SAP in K , for each d ∈ D ,there are embeddings f d , g d and G d ∈ K so that the following diagram commutes, C d f d (cid:33) (cid:33) A d i (cid:62) (cid:62) j (cid:32) (cid:32) G d B d g d (cid:61) (cid:61) where i and j are the induced maps, so that f d ( C d ) ∩ g d ( B b ) = ( f d ◦ i )( A d ). Sincethe language is relational, HP implies that we may take G d = f d ( C d ) ∪ g d ( D d ).Moreover, we may choose f d and g d so that they are the same functions for all d ∈ D on the underlying sets C and B respectively. Call these functions f and g . Let G be the underlying set of G d for some (all) d ∈ B . Now define a structure ( G, E ∪ F )so that for all d ∈ D = E ∩ F , G d is as above, if a ∈ E \ F , G a is some structurein K extending g ( B a ) and, likewise, if a ∈ F \ E , G a is some structure extending f ( C a ). The functions f and g extend to embeddings f : ( C, F ) → ( G, E ∪ F )and g : ( B, E ) → ( G, E ∪ F ) so that f and g are both inclusions on parameters. N MODEL-THEORETIC TREE PROPERTIES 25 By construction, it is clear that f i = gj . Moreover, f i ( A ) = f ( C ) ∩ g ( B ) and f i ( D ) = f ( E ) ∩ g ( F ), which establishes SAP in K pfc . (cid:3) As K pfc is a Fra¨ıss´e class, there is a unique countable ultrahomogeneous L pfc -structure with age K pfc . Let T pfc denote its theory. By Fra¨ıss´e theory, this theoryeliminates quantifiers and is ℵ -categorical. Lemma 6.12. Suppose ( A, B ) | = T pfc . Then, for all b ∈ B , A b | = T .Proof. Since the property that for all b ∈ B , A b | = T is an elementary property,it suffices to check this when ( A, B ) is the unique countable model of T pfc . If d, e ∈ A b satisfy tp L ( d ) = tp L ( e ) then, by quantifier-elimination, it is easy to checktp L pfc ( b, d ) = tp L pfc ( b, e ) and ultrahomogeneity of ( A, B ) implies there is an L pfc -automorphism of ( A, B ) fixing b and taking d to e . The induced L -automorphismof A b witnesses that A b is ultrahomogeneous. By Fra¨ıss´e theory there is up toisomorphism a unique countable ultrahomogeneous L -structure with age K so A b is isomorphic to a model of T , so A b | = T . (cid:3) Suppose M = ( A , B ) is a monster model of T pfc . Given a formula ϕ ∈ L and aparameter p ∈ B , define ϕ p ∈ L pfc to be the formula obtained by replacing eachoccurrence of R i by R ip and giving the objects their eponymous interpretations in A p – formally, this defines ϕ p for atomic ϕ and then the full definition follows byinduction on the complexity of the formulas. If C ⊆ A is a set of objects and q isan L -type over C (considered as a subset of A p ), we define the type q p by q p = { ϕ p : ϕ ∈ q } . Lemma 6.13. Suppose { p i : i < α } ⊆ B is a collection of distinct parametersand q i : i < α ) is a sequence of non-algebraic L -types over C ⊆ A (possibly withrepetition), where q i is considered as a type in A p i . Then the L pfc -type (cid:83) i<α q ip i isconsistent.Proof. By compactness, it suffices to consider the case where α < ω and when the q i are all finite types. Hence, we simply have to show M | = ( ∃ x ) (cid:94) i<α q ip i ( x ) . Moreover, by quantifier-elimination in T , we may assume that each q i is quantifier-free. For each i < α , let C i ∈ Age( A p i ) the finite substructure generated by theelements of C mentioned in all of the q i . So, the underlying set of each C i is thesame, although the interpretations of the relations may differ. Given any i < α , weknow that A p i | = ( ∃ x ) (cid:94) q ip i ( x )so there is D i ∈ Age( A p i ) containing a witness d i to the above existential formula.By non-algebraicity of each type, we may assume that d i (cid:54)∈ C i and, by HP, that D i = C ∪ { d i } .Now define an L pfc -structure E with underlying set of objects C ∪ {∗} where ∗ is some new element and its parameters are { p i : i < α } , and the relations areinterpreted so that for each i < α , the map is the identity on C and sends d i (cid:55)→ ∗ isan isomorphism of L -structures from D i to E p i . It is clear that E ∈ K pfc so there is a copy F isomorphic over C ∪ { p i : i < α } to it in Age( M ). Now F | = ( ∃ x ) (cid:94) i<α q ip i ( x )and hence this is satisfied in M , so we’re done. (cid:3) Lemma 6.14. Suppose A, B, C ⊆ A are small sets of objects, F ⊆ B is a small setof parameters, A ∩ B ⊆ C , and b , b ∈ B satisfy b ≡ CF b . Then there is some b ∈ B so that b ≡ ACF b and b ≡ BCF b (all in L pfc ).Proof. Given a set D ⊆ A and p ∈ B , recall that we write (cid:104) D (cid:105) p for the L -substructure of A p with underlying set D . By compactness, it suffices to prove thelemma when A, B, C, and F are finite. By quantifier-elimination, demanding some b ∈ B so that b ≡ AC b and b ≡ BC b is equivalent to asking that (cid:104) AC (cid:105) b ∼ = (cid:104) AC (cid:105) b and (cid:104) BC (cid:105) b ∼ = (cid:104) AC (cid:105) b . Now, as b ≡ C b , (cid:104) C (cid:105) b may be identified with (cid:104) C (cid:105) b .We may view C, (cid:104) AC (cid:105) b , and (cid:104) BC (cid:105) b as elements of K . In K , we have inclusions i : C → (cid:104) AC (cid:105) b and j : C → (cid:104) BC (cid:105) b , so by SAP, there are embeddings f, g and a D ∈ K so that the following diagram commutes (cid:104) AC (cid:105) b f (cid:35) (cid:35) C i (cid:59) (cid:59) j (cid:35) (cid:35) D (cid:104) BC (cid:105) b g (cid:59) (cid:59) where f ( AC ) ∩ g ( BC ) = C . By HP, D may be taken to have f ( AC ) ∪ g ( BC ) as itsdomain. Since A ∩ B ⊆ C , D is isomorphic over C to an L -structure with underlyingset A ∪ B ∪ C , so we may assume that f and g are both inclusions. Let b ∗ denotesome new parameter element outside of F and define a structure with parameterset { b ∗ , b , b } ∪ F and A ∪ B ∪ C as its set of objects so that (cid:104) ABC (cid:105) b ∗ ∼ = D . Thisclearly defines a structure in K pfc . In the substructure with only A ∪ C as the setof objects, there is an automorphism fixing F taking b ∗ to b . This shows that b ∗ ≡ ACF b and a symmetric argument shows b ∗ ≡ BCF b . It follows that we canfind such a b ∗ in B . (cid:3) Towards proving an independence theorem for T pfc , we will define a notion ofindependence for parameterized structures. Definition 6.15. ( | (cid:94) pfc )(1) Suppose p ∈ B is a parameter. Suppose A, B, C ⊆ A . We define | (cid:94) p by A p | (cid:94) C B in M ⇐⇒ A | (cid:94) C B in A p , where the undecorated | (cid:94) on the right-hand side denotes the usual non-forking independence – i.e. tp( A/BC ) does not fork over C .(2) If A, B, C ⊆ A and D, E, F ⊆ B , we define | (cid:94) pfc by A, D pfc | (cid:94) C,F B, E ⇐⇒ D ∩ E ⊆ F, and for all p ∈ F, A p | (cid:94) C B. N MODEL-THEORETIC TREE PROPERTIES 27 Proposition 6.16. Assume T is a simple theory. Suppose A, B ⊆ A are small setsof objects and D, E ⊆ B are small sets of parameters and M = ( C, F ) is a smallmodel of T pfc satisfying A, D pfc | (cid:94) C,F B, E Suppose moreover that a , a are tuples from A and b , b are tuples from B satis-fying a , b | (cid:94) pfc CF A, D , a , b | (cid:94) pfc C,F B, E and a , b ≡ CF a , b . Then there are a from A and b from B so that a, b ≡ ACDF a , b and a, b ≡ BCEF a , b .Proof. First, we solve the amalgamation problem for objects. Without loss ofgenerality, D, E, F are pairwise disjoint. By Lemma 6.12, we know that for each p ∈ F , C p is a model of T . By definition of | (cid:94) pfc , we know that in A p , we have A | (cid:94) pC B , a | (cid:94) pC A and a | (cid:94) pC B . As T is simple, the independence theorem overa model implies that there is some tuple a p in A p such that a p ≡ LAC a , a p ≡ LBC a and a p | (cid:94) pC AB . For each p ∈ F , let q p ( x ) = tp L ( a p /ABC ) considered as an L -type in A p . By Lemma 6.13, denoting the relativization of q p to the parametrizedlanguage with respect to p by q pp , we know that the type (cid:83) p ∈ F q pp is consistent. Let a be a realization. Then a ≡ AC a and a ≡ BC a in A p for all p ∈ F so a ≡ ACF a and a ≡ BCF a .Now we solve the problem for parameters. First assume that b , b are single-tons in B . Without loss of generality b , b / ∈ F (as otherwise they are equal byassumption, and there is nothing to do). By quantifier-elimination, we need some b (cid:54)∈ D ∪ E ∪ F so that (cid:104) aAC (cid:105) b ∼ = (cid:104) a AC (cid:105) b and (cid:104) aBC (cid:105) b ∼ = (cid:104) a BC (cid:105) b . First, find b ≡ ACF b and b ≡ BCF b outside of D ∪ E ∪ F so that (cid:104) aAC (cid:105) b ∼ = (cid:104) a AC (cid:105) b and (cid:104) aBC (cid:105) b ∼ = (cid:104) a BC (cid:105) b . So ab ≡ ACF a b and ab ≡ BCF a b . Now b ≡ aCF b and aAC ∩ aBC ⊆ aC , so Lemma 6.14 applies and we can find a b so that (cid:104) aAC (cid:105) b ∼ = (cid:104) aAC (cid:105) b and (cid:104) aBC (cid:105) b ∼ = (cid:104) aBC (cid:105) b , and we can take this b to be outsideof D ∪ E ∪ F . Now as b (cid:54)∈ D ∪ E ∪ F , we have ab ≡ ACDF a b and ab ≡ BCEF a b .Now let b = ( b ,i : i < k ) , b = ( b ,i : i < k ) be arbitrary tuples from B .Without loss of generality, all of the elements in { b t,i : i < k } are pairwise-distinct,for t ∈ { , } . Let S t = { i < k : b t,i / ∈ F } for t ∈ { , } , note that S = S = S as b ≡ F b . Repeatedly applying the argument above for singletons, we can find pairwise distinct b (cid:48) i for i ∈ S such that a, b (cid:48) i ≡ ACDF a , b ,i and a, b (cid:48) i ≡ BCEF a , b ,i for all i ∈ S . Let b ∗ = ( b ∗ i : i < k ) be defined by taking b ∗ i = b ,i = b ,i for all i / ∈ S and b ∗ i = b (cid:48) i for all i ∈ S . As there are no relations in the language involving morethan one element from the parameter sort except for the equality, it follows that a, b ∗ ≡ ACDF a , b and a, b ∗ ≡ BCEF a , b — as wanted. (cid:3) Theorem 6.17. Assume T is simple. Then | (cid:94) pfc is an Aut ( M ) -invariant inde-pendence relation on small subsets of the monster M | = T pfc such that it satisfies,for an arbitrary M | = T pfc :(1) strong finite character: if a (cid:54) | (cid:94) pfc M b , then there is a formula ϕ ( x, b, m ) ∈ tp ( a/bM ) such that for any a (cid:48) | = ϕ ( x, b, m ) , a (cid:48) (cid:54) | (cid:94) pfc M b ;(2) existence over models: M | = T pfc implies a | (cid:94) pfc M M for any a ;(3) monotonicity: aa (cid:48) | (cid:94) pfc M bb (cid:48) = ⇒ a | (cid:94) pfc M b ;(4) symmetry: a | (cid:94) pfc M b ⇐⇒ b | (cid:94) pfc M a ; (5) independent amalgamation: c | (cid:94) pfc M c , b | (cid:94) pfc M c , b | (cid:94) pfc M c , b ≡ M b implies there exists b with b ≡ c M b , b ≡ c M b .Proof. Automorphism invariance and (1)-(4) are immediate from the definition of | (cid:94) pfc , using that T is simple and hence non-forking independence satisfies all theseproperties; (5) was proven in Proposition 6.16. (cid:3) Corollary 6.18. Suppose T is a simple theory which is the theory of a Fra¨ıss´e limitof a Fra¨ıss´e class K satisfying SAP. Then T pfc is NSOP . Moreover, if the D -rankof T is ≥ , then T pfc is not simple.Proof. By Proposition 5.8, T pfc is NSOP , as | (cid:94) pfc gives an independence relationsatisfying all the hypotheses. So now we prove that T pfc is not simple, under theassumption that the D -rank of T is ≥ 2. This assumption implies that there is an L -formula ϕ ( x ; y ) and an indiscernible sequence ( a i ) i<ω so that { ϕ ( x ; a i ) : i < ω } is k -inconsistent for some k and the set defined by ϕ ( x ; a i ) is infinite. Let M | = T be some model containing the sequence ( a i ) i<ω . Construct an L pfc -structure N with domain ω (cid:116) M and relations interpreted so that N | = R i ( b ) ⇐⇒ M | = R ( b )for each tuple b ∈ M , every i < ω , and relation symbol R of L . Extend N to˜ N | = T pfc . Let ψ ( x ; y, z ) be the formula ϕ z ( x ; y ) and define an array ( b ij ) i,j<ω by b ij = ( a j , i ) ∈ M × ω ⊂ ˜ N . Using Lemma 6.13, it is easy to check thatfor all f : ω → ω , (cid:83) i<ω { ψ ( x ; b if ( i ) ) } is consistent. Also { ψ ( x ; b ij ) : j < ω } is k -inconsistent for all i so ψ witnesses TP . (cid:3) Remark . For the above argument to work, we used that the formula witnessingdividing was non-algebraic — this fails in many natural examples (e.g. the randomgraph). However, given an L -structure M , define the imaginary cover of M asfollows: let L (cid:48) be the language L together with a new binary relation symbol E foran equivalence relation, and let ˜ M be the L (cid:48) -structure obtained by replacing eachelement of M with an infinite E -class and defining the relations of L on ˜ M on thecorresponding E -classes. Now it is easy to check that Age( ˜ M ) has SAP, the theoryof ˜ M is simple of D -rank at least 2. Corollary 6.20. T ∗ feq is NSOP .Proof. The theory T of an equivalence relation with infinitely many infinite classesis a stable theory, obtained as the Fra¨ıss´e limit of all finite models of the theory ofan equivalence relation. This class has no algebraicity, so it satisfies SAP. T pfc isexactly T ∗ feq , so it is NSOP . (cid:3) This result was claimed in [22], but the proof is apparently incorrect due toan illegitimate use of tree-indiscernibles. See the footnote on [13, p. 22] for adiscussion.6.4. Theories approximated by simple theories. In her thesis [13], GwynethHarrison-Shermoen considers theories that have a model approximated by a di-rected system H of homogeneous substructures, each of which has a simple theory.She proves that such theories carry an invariant independence notion | (cid:94) lim satis-fying strong finite character, monotonicity, symmetry, and existence over a model(existence over a model is implied by Claim 3.3.4 in [13]). Finally, she observes N MODEL-THEORETIC TREE PROPERTIES 29 that if non-forking independence | (cid:94) f satisfies the independence theorem over alge-braically closed sets for each model in H , then so does | (cid:94) lim for the approximatedtheory. Hence, we obtain the following: Corollary 6.21. Suppose T is a theory approximated, in the sense of Harrison-Shermoen, by a directed system of structures each with a simple theory in which | (cid:94) f satisfies the independence theorem over algebraically closed sets. Then T isNSOP . Lemmas on preservation of indiscernibility Lemma 7.1. Suppose η , . . . , η l − , ν , . . . , ν l − are elements of ω <ω . Let η and ν denote enumerations of the ∧ -closures of { η i : i < l } and { ν i : i < l } respectively.Then if qftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) , then qftp L s ( η ) = qftp L s ( ν ) . Proof. Easy. See Remark 3.2 from [17] (cid:3) Lemma 7.2. Let η , . . . , η l − , ν , . . . , ν l − ∈ ω <ω be such thatqftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) . Suppose i < l and η (cid:67) η i , ν (cid:67) ν i with l ( η ) = l ( ν ) . Then, setting η l = η and ν l = ν ,we have qftp L s ( η , . . . , η l ) = qftp L s ( ν , . . . , ν l ) . Proof. Without loss of generality, we may take { η i : i < l } and { ν i : i < l } to be ∧ -closed, by the previous lemma. Then { η i : i < l + 1 } and { ν i : i < l + 1 } are also ∧ -closed. So we need only to check that for any j, j (cid:48) < l + 1(1) η j (cid:67) η j (cid:48) ⇐⇒ ν j ⇐⇒ ν (cid:48) j (2) η j < lex η j (cid:48) ⇐⇒ ν j < lex ν j (cid:48) We have 3 cases. Case 1: j, j (cid:48) < l .(1) and (2) follow by assumption. Case 2: j < l and j (cid:48) = lη j (cid:67) η l ⇐⇒ η j (cid:67) η i and l ( η j ) ≤ l ( η l ) ⇐⇒ η j (cid:67) η i ∧ (cid:95) k Let ( a η ) η ∈ ω <ω be an s -indiscernible tree. If ( a (cid:48) η ) η ∈ ω <ω is the k -foldwidening of ( a η ) η ∈ ω <ω at level n , then ( a (cid:48) η ) η ∈ ω <ω is also s -indiscernible.Proof. Pick η , . . . , η l − and ν , . . . , ν l − in ω <ω so thatqftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) . By Lemma 7.2, we may assume that { η i : i < l } and { ν i : i < l } are both ∧ -closedand closed under initial segment. Moreover, we may assume that these elementshave been enumerated so that for some m ≤ l , l ( η i ) , l ( ν i ) < n if and only if i ≥ m .So for each i < m , we may write η i = µ i (cid:95) α i (cid:95) ξ i ν i = υ i (cid:95) β i (cid:95) ρ i , N MODEL-THEORETIC TREE PROPERTIES 31 where µ i , υ i ∈ ω n − , α i , β i ∈ ω , and ξ i , ρ i ∈ ω <ω . For each i < m , let η i = ( µ i (cid:95) ( kα i ) (cid:95) ξ i , µ i (cid:95) ( kα i + 1) (cid:95) ξ i , . . . , µ i (cid:95) ( kα i + k − (cid:95) ξ i ) ν i = ( υ i (cid:95) ( kβ i ) (cid:95) ρ i , υ i (cid:95) ( kβ i + 1) (cid:95) ρ i , . . . , υ i (cid:95) ( kβ i + k − (cid:95) ρ i ) . and for m ≤ i < l , let η i = η i , ν i = ν i . Now we must show thatqftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) . It is clear that the sets (cid:83) i 1. In the firstcase, the meet is enumerated in one of the tuples because our initial set of tupleswas ∧ -closed, in the second case because it was taken to be closed under initialsegment. To check equality of the quantifier-free types, we have 3 cases: Case 1: i, i (cid:48) ≥ m Follows by assumption, as for any i ≥ m , η i = η i and ν i = ν i . Case 2: i ≥ m , i (cid:48) < m and j < kη i (cid:67) ( η i (cid:48) ) j ⇐⇒ ν i (cid:67) ( ν i (cid:48) ) j η i < lex ( η i (cid:48) ) j ⇐⇒ ν i < lex ( η i (cid:48) ) j ( η i (cid:48) ) j < lex η i ⇐⇒ ( ν i (cid:48) ) j < lex ν i Case 3: i, i (cid:48) < m and j, j (cid:48) < k ( η i ) j (cid:67) ( η i (cid:48) ) j (cid:48) ⇐⇒ η i (cid:67) η i (cid:48) and j = j (cid:48) ⇐⇒ ν i (cid:67) ν i (cid:48) and j = j (cid:48) ⇐⇒ ( ν i ) j (cid:67) ( ν i (cid:48) ) j (cid:48) ( η i ) j < lex ( η i (cid:48) ) j (cid:48) ⇐⇒ ( η i < lex η j and ( l ( η i ∧ η j ) < n or j = j (cid:48) )) or( l ( η i ∧ η i (cid:48) ) ≥ n and j < j (cid:48) ) ⇐⇒ ( ν i < lex ν j and ( l ( ν i ∧ ν j ) < n or j = j (cid:48) )) or( l ( ν i ∧ ν i (cid:48) ) ≥ n and j < j (cid:48) ) ⇐⇒ ( ν i ) j < lex ( ν i (cid:48) ) j (cid:48) . (cid:3) Lemma 7.4. Let ( a η ) η ∈ ω <ω be an s -indiscernible tree. If ( a (cid:48)(cid:48) η ) η ∈ ω <ω is the k -foldstretch of ( a η ) η ∈ ω <ω at level n , then ( a (cid:48)(cid:48) η ) η ∈ ω <ω is also s -indiscernible.Proof. Given η ∈ ω <ω , let η = η if l ( η ) < n ( η, η (cid:95) , . . . , η (cid:95) k − ) if l ( η ) = nν (cid:95) k − (cid:95) ξ if η = ν (cid:95) ξ, with ν ∈ ω n , ξ (cid:54) = ∅ Pick η , . . . , η l − , ν , . . . , ν l − ∈ ω <ω so thatqftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) , and, without loss of generality, we may suppose { η i : i < l } and { ν i : i < l } areboth ∧ -closed. We must show thatqftp L s ( η , . . . , η l − ) = qftp L s ( ν , . . . , ν l − ) . Assume that { η i : i < l } is ordered so that i < m if and only if l ( η i ) = n , andsimilarly for { ν i : i < l } . Clearly { η i : i < l } and { ν i : i < l } are also ∧ -closed,so we have to check that the two sequences of tuples have the same quantifier typewith respect to the relations < lex and (cid:67) . We’ll show this by considering the variouscases: Case 1 : i, i (cid:48) ≥ m . Then η i (cid:67) η i (cid:48) ⇐⇒ η i (cid:67) η i (cid:48) ⇐⇒ ν i (cid:67) ν i (cid:48) ⇐⇒ ν i (cid:67) ν i (cid:48) η i < lex η i ⇐⇒ η i < lex η i (cid:48) ⇐⇒ ν i < lex ν i (cid:48) ⇐⇒ ν i < lex ν i (cid:48) . Case 2 : i, i (cid:48) < m and j, j (cid:48) < k . Then( η i ) j (cid:67) ( η i (cid:48) ) j (cid:48) ⇐⇒ ( η i = η i (cid:48) ) ∧ ( j < j (cid:48) ) ⇐⇒ ( ν i = ν i (cid:48) ) ∧ ( j < j (cid:48) ) ⇐⇒ ( ν i ) j (cid:67) ( ν ) j (cid:48) ( η i ) j < lex ( η i (cid:48) ) j (cid:48) ⇐⇒ η i < lex η i (cid:48) ∨ ( ν i = ν i (cid:48) ∧ j < j (cid:48) ) ⇐⇒ ν i < lex ν i (cid:48) ∨ ( ν i = ν i (cid:48) ∧ j < j (cid:48) ) ⇐⇒ ( ν i ) j < lex ( ν i (cid:48) ) j (cid:48) . Case 3 : i < m , i (cid:48) ≥ m , j < k .( η i ) j (cid:67) η i (cid:48) ⇐⇒ η i (cid:67) η i (cid:48) ⇐⇒ ν i (cid:67) ν i (cid:48) ⇐⇒ ( ν i ) j (cid:67) ν i η i (cid:48) (cid:67) ( η i ) j ⇐⇒ η i (cid:48) (cid:67) η i ⇐⇒ ν i (cid:48) (cid:67) ν i ⇐⇒ ( ν i (cid:48) ) j (cid:67) ν i ( η i ) j < lex η i (cid:48) ⇐⇒ η i < lex η i (cid:48) ⇐⇒ ν i < lex ν i (cid:48) ⇐⇒ ( ν i ) j < lex ν i (cid:48) η i (cid:48) < lex ( η i ) j ⇐⇒ ν i (cid:48) < lex ν i ⇐⇒ ν i (cid:48) < lex ν i ⇐⇒ ν i (cid:48) < lex ( ν i ) j . (cid:3) Lemma 7.5. (1) Each tuple a ( n ) η may be enumerated as ( a ν(cid:95)η : ν ∈ n ) (2) If ( a η ) η ∈ <κ is strongly indiscernible, then for all n , the n -fold fattening ( a ( n ) η ) η ∈ <κ is strongly indiscernible over C n Proof. (1) This is trivial for n = 0. Then if true for n , we have a ( n +1) η = ( a ( n )0 (cid:95)η , a ( n )1 (cid:95)η ) = (( a ν(cid:95) (cid:95)η : ν ∈ n ) , ( a ν(cid:95) (cid:95)η : ν ∈ n )) = ( a ξ(cid:95)η : ξ ∈ n +1 ) . N MODEL-THEORETIC TREE PROPERTIES 33 (2) By (1) we have a ( n +1) η = ( a µ(cid:95)η : µ ∈ n ). Let µ = ( µ ∈ ≤ n ). In order to showindiscernibility over C n have to show that if η , . . . , η k − , ν , . . . , ν k − ∈ <ω andqftp L ( η , . . . , η k − ) = qftp L ( ν , . . . , ν k − )then qftp L ( µ, ( a µ(cid:95)η : µ ∈ n ) , . . . , ( a µ(cid:95)η k − : µ ∈ n )) is equal to qftp L ( µ, ( a µ(cid:95)ν : µ ∈ n ) , . . . , ( a µ(cid:95)ν k − : µ ∈ n )). To this end, we may assume { η , . . . , η k − } and { ν , . . . , ν k − } are meet-closed. Then 2 ≤ n ∪{ µ (cid:95) η i : µ ∈ n , i < k } and 2 ≤ n ∪{ µ (cid:95)ν i : µ ∈ n , i < k } is also meet-closed and we just have to check that the tuples inthe above equation have the same time with respect to the language L t = { (cid:67) , < lex } .Choose ξ , ξ from the tuple ( µ, ( a µ(cid:95)η : µ ∈ n ) , . . . , ( a µ(cid:95)η k − : µ ∈ n )) and ρ , ρ from ( µ, ( a µ(cid:95)ν : µ ∈ n ) , . . . , ( a µ(cid:95)ν k − : µ ∈ n )) so that ξ i sits in the same po-sition in the enumeration of the tuple as ρ i for i = 0 , 1. Now, we must show that ξ < lex ξ if and only if ρ < lex ρ and ξ (cid:69) ξ if and only if ρ (cid:69) ρ . Choosearbitrary µ , µ ∈ ≤ n , η i , η j , ν i , ν j . Case 1 : l ( µ ) = l ( µ ) = n , ξ = µ (cid:95) η i , ξ = µ (cid:95) η j , and hence ρ = µ (cid:95) ν i and ρ = µ (cid:95) ν j . µ (cid:95) η i (cid:69) µ (cid:95) η j ⇐⇒ µ = µ ∧ η i (cid:69) η j ⇐⇒ µ = µ ∧ ν i (cid:69) ν j ⇐⇒ µ (cid:95) ν i (cid:67) µ (cid:95) ν j µ (cid:95) η i < lex µ (cid:95) η j ⇐⇒ µ < lex µ ∨ ( µ = µ ∧ η i < lex η j ) ⇐⇒ µ < lex µ ∨ ( µ = µ ∧ ν i < lex ν i (cid:48) ) ⇐⇒ µ (cid:95) ν i < lex µ (cid:95) ν j Case 2 : ξ = µ , ξ = µ , ρ = µ , and ρ = µ .Clear. Case 3 : l ( µ ) = n , ξ = µ (cid:95) η i , ξ = µ , ρ = µ (cid:95) ν i , ρ = µ .It is never the case that µ (cid:95) η i (cid:67) µ or µ (cid:95) ν i (cid:67) µ so it suffices to check < lex : µ (cid:95) η i < lex µ ⇐⇒ µ < lex µ ⇐⇒ µ (cid:95) ν i < lex µ . Case 4 : l ( µ ) = n , ξ = µ , ξ = µ (cid:95) ν j , ρ = µ , ρ = µ (cid:95) ν j . µ (cid:69) µ (cid:95) η j ⇐⇒ µ (cid:69) µ ⇐⇒ µ (cid:69) µ (cid:95) ν j µ ≤ lex µ (cid:95) η j ⇐⇒ µ ≤ lex µ ⇐⇒ µ ≤ lex µ (cid:95) ν j (cid:3) Lemma 7.6. If ( a η ) η ∈ <ω is strongly indiscernible, then for all natural numbers k ≥ , the k -fold elongation ( a (cid:48) η ) η ∈ <ω of ( a η ) η ∈ <ω is also strongly indiscernible.Proof. Given η ∈ <ω , with l ( η ) = n , we defined ˜ η ∈ <ω to be the element withlength k ( l ( η ) − 1) + 1 defined by˜ η ( i ) = (cid:26) η ( i/k ) if k | i As the k -fold elongation of ( a η ) η ∈ <ω is defined to be the tree ( b η ) η ∈ <ω where b η = ( a ˜ η , a ˜ η(cid:95) , . . . , a ˜ η(cid:95) k − ) . Write η for the tuple (˜ η, ˜ η (cid:95) , . . . , ˜ η (cid:95) k − ). We are reduced to showing that if η , . . . , η l − , ν , . . . , ν l − are elements of 2 <ω so thatqftp L ( η , . . . , η l − ) = qftp L ( ν , . . . , ν l − )then qftp L ( η , . . . , η l − ) = qftp L ( ν , . . . , ν l − ) . We may assume that { η i : i < l } and { ν i : i < l } are both ∧ -closed, from which itfollows that { η i : i < l } and { ν i : i < l } are both ∧ -closed. So we must check that( η i : i < l ) and ( ν i : i < l ) have the same quantifier-free type with respect to thelanguage L t = (cid:104) (cid:69) , < lex (cid:105) . We note˜ η i (cid:95) l (cid:69) ˜ η j (cid:95) l (cid:48) ⇐⇒ ˜ η i (cid:67) ˜ η j ∨ (˜ η i = ˜ η j ∧ l ≤ l (cid:48) ) ⇐⇒ η i (cid:67) η j ∨ ( η i = η j ∧ l ≤ l (cid:48) ) ⇐⇒ ν i (cid:67) ν j ∨ ( ν i = ν j ∧ l ≤ l (cid:48) ) ⇐⇒ ˜ ν i (cid:67) ˜ ν j ∨ (˜ ν i = ˜ ν j ∧ l ≤ l (cid:48) ) ⇐⇒ ˜ ν i (cid:95) l (cid:67) ˜ ν j (cid:95) l (cid:48) ˜ η i (cid:95) l < lex ˜ η j (cid:95) l (cid:48) ⇐⇒ ˜ η i < lex ˜ η j ∨ (˜ η i = ˜ η j ∧ l < l (cid:48) ) ⇐⇒ η i < lex η j ∨ ( η i = η j ∧ l < l (cid:48) ) ⇐⇒ ν i < lex ν j ∨ ( ν i = ν j ∧ l < l (cid:48) ) ⇐⇒ ˜ ν i < lex ˜ ν j ∨ (˜ ν i = ˜ ν j ∧ l < l (cid:48) ) ⇐⇒ ˜ ν i (cid:95) l < lex ˜ ν j (cid:95) l (cid:48) . (cid:3) Lemma 7.7. Suppose ( a η ) η ∈ <ω is a strongly indiscernible tree over C .(1) Define a function h : 2 <ω → <ω by h ( ∅ ) = ∅ and h ( η ) = h ( ν ) (cid:95) (cid:95) (cid:104) i (cid:105) whenever η = ν (cid:95) (cid:104) i (cid:105) . Then ( a h ( η ) ) η ∈ <ω is strongly indiscernible over C .(2) For each n , define a map h n : 2 <ω → <ω by h n ( η ) = (cid:26) h ( η ) if l ( η ) ≤ nh ( ν ) (cid:95) ξ if η = ν (cid:95) ξ, l ( ν ) = n. Then ( a h n ( η ) ) η ∈ <ω is strongly indiscernible over C .Proof. (1) At the outset, we note that η (cid:69) ν ⇐⇒ h ( η ) (cid:69) h ( ν ) and η < lex ν ⇐⇒ h ( η ) < lex h ( ν ). The only difficulty arises from ∧ which is not preserved by h ,because if η ⊥ ν and η ∧ ν = ξ then h ( η ) ∧ h ( ν ) = h ( ξ ) (cid:95) η, ν are finite tuples from 2 <ω with qftp L ( η ) =qftp L ( ν ) then qftp L ( h ( η )) = qftp L ( h ( ν )). Given such η, ν , it is clear that ifqftp L ( h ( η )) (cid:54) = qftp L ( h ( ν )) then qftp L ( h ( η (cid:48) )) (cid:54) = qftp L ( h ( ν (cid:48) )) where η (cid:48) and ν (cid:48) are the ∧ -closures of η and ν respectively. So we may assume η and ν are ∧ -closed.We may assume that the tuple η = (cid:104) η i : i < k (cid:105) is enumerated so that for some l ≤ k ,if i < l , then there are η j ⊥ η j (cid:48) so that η j ∧ η j (cid:48) = η i . It follows that the ∧ -closureof h ( η ) may be enumerated as (cid:104) h ( η i ) : i < k (cid:105) (cid:95) (cid:104) h ( η i ) (cid:95) i < l (cid:105) , and, likewise, N MODEL-THEORETIC TREE PROPERTIES 35 the ∧ -closure of h ( ν ) can be enumerated as (cid:104) h ( ν i ) : i < k (cid:105) (cid:95) (cid:104) h ( ν i ) (cid:95) i < l (cid:105) .Now we note that, by definition of h , if i, j < kh ( η i ) (cid:67) h ( η j ) (cid:95) ⇐⇒ h ( η i ) (cid:95) (cid:67) h ( η j ) ⇐⇒ h ( η i ) (cid:95) (cid:67) h ( η j ) (cid:95) ⇐⇒ h ( η i ) (cid:67) h ( η j ) h ( η i ) < lex h ( η j ) (cid:95) ⇐⇒ h ( η i ) (cid:95) < lex h ( η j ) ⇐⇒ h ( η i ) (cid:95) < lex h ( η j ) (cid:95) ⇐⇒ h ( η i ) < lex h ( η j )And similarly for ν i , ν j . As h respects (cid:67) and < lex , and qftp L ( η ) = qftp L ( ν ), itfollows that qftp L ( h ( η )) = qftp L ( h ( ν )).(2) is entirely similar. (cid:3) References [1] Hans Adler. Strong theories, burden, and weight. Unpublished, , 2007.[2] Hans Adler. 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