FORKING AND DIVIDING IN HENSON GRAPHS

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1 FORKING AND DIVIDING IN HENSON GRAPHS GABRIEL ONANT arxiv: v1 [math.lo] 8 Jan 2014 Abstract. For n 3, define T n to be the theory of the generic K n-free graph, where K n is the complete graph on n vertices. We prove a graph theoretic characterization of dividing in T n, and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, T n provides a counterexample to a recent question of hernikov and Kaplan. 1. Introduction lassification in model theory, beginning with stability theory, is strongly fueled by the study of abstract notions of independence, the frontrunners of which are forking and dividing. These notions have proved useful in the abstract treatment of independence and dimension in the stable setting, and initiated a quest to understand when they are useful in the unstable context. Significant success was achieved in the class of simple theories (see [6]). Meaningful results have also been found for NIP theories and, more generally, NTP 2 theories, which include both simple and NIP. A notable example is the following recent result from [3]. Theorem. Suppose M is a sufficiently saturated monster model of an NTP 2 theory. Given M, the following are equivalent. (i) A partial type forks over if and only if it divides over. (ii) is an extension base for nonforking, i.e. if π( x) is a partial type with parameters from, then π( x) does not fork over. In general, if condition (i) holds for a set, then condition (ii) does as well. In fact, condition (ii) should be thought of as the minimal requirement for nonforking to be meaningful for types over. In particular, if is not an extension base for nonforking, then there are types with no nonforking extensions. There are few examples where condition (ii) fails, and most of them do so by exploiting some kind of circular ordering (see e.g [9, Exercise 7.1.6]). On the other hand, condition (i) is, a priori, harder to achieve. It is useful because it allows us to ignore the subtlety of forking versus dividing. However, in every textbook example where condition (i) fails, it is because condition (ii) also fails. This leads to the natural question, which is asked in [3], of whether the result above extends to classes of theories other than NTP 2. In this paper, we give an example of an NSOP 4 theory in which condition (ii) holds for all sets, but condition (i) fails. We will consider forking and dividing in the theory of a well-known structure: the generic K n -free graph, also known as the Henson graph, a theory with TP 2 and NSOP 4. Our main goal is to characterize forking and dividing in the theory of the Henson graph. We will show that dividing independence has a meaningful graphtheoretic interpretation, and has something to say about the combinatorics of the 1

2 2 GABRIEL ONANT structure. Using this characterization, we will show that despite the complexity of the theory, forking and dividing are the same for complete types. As a consequence, every set is an extension base for nonforking, and so nonforking/nondividing extensions always exist. On the other hand, we will show that there are formulas which fork, but do not divide. Acknowledgements. I would like to thank Lynn Scow, John Baldwin, Artem hernikov, Dave Marker, aroline Terry, and Phil Wesolek for their part in the development of this project. 2. Model Theoretic Preliminaries This section contains the definitions and basic facts concerning forking and dividing. We first specify some conventions that will be maintained throughout the paper. If T is a complete first order theory and M is a monster model of T, we write A M to mean that A is a small subset of M, i.e. A M and M is A + -saturated. We use the letters a,b,c,... to denote singletons, and ā, b, c,... to denote tuples (of possibly infinite length). Definition 2.1. Suppose M, π( x,ȳ) is a partial type with parameters from, and b M. (1) π( x, b) divides over if there is a -indiscernible sequence ( b l ) l<ω, with b0 = b, such that l<ω π( x, b l ) is inconsistent. (2) π( x, b) forks over if there is some D b such that if p S n (D) extends π( x, b) then p divides over. (3) A formula ϕ( x, b) forks (resp. divides) over if {ϕ( x, b)} forks (resp. divides) over. The following basic facts can be found in [9]. Proposition 2.2. Let M. (a) If a complete type forks (resp. divides) over then it contains some formula that forks (resp. divides) over. (b) If π( x) is a consistent type over then π( x) does not divide over. Nondividing and nonforking are used to define ternary relations on small subsets of M, given by (1) A d B if and only if tp(a/b) does not divide over, (2) A f B if and only if tp(a/b) does not fork over. These relations were originally defined to abstractly capture notions of independence and dimension in stable theories, and have been found to still be meaning ful in more complicated theories as well. In particular, we will consider the interpretation of these notions in the unstable theories of certain homogeneous graphs. 3. Graphs Recall that a countable graph G is universal if any countable graph is isomorphic to aninduced subgraphofg; and Gis homogenousif anygraphisomorphism between finite subsets of G extends to an automorphism of G. The canonical example of such a graph is the countable random graph, i.e. the Fraïssé limit of the class of finite graphs. In [5], a new family of countable homogenous graphs was introduced: the generic K n -free graphs, for n 3, which are often

3 FORKING AND DIVIDING IN HENSON GRAPHS 3 called the Henson graphs. For a particular n 3, there is a unique such graph up to isomorphism. Definition 3.1. Fix n 3 and let K n be the complete graph on n vertices. The generic K n -free graph, H n, is the unique countable graph such that (i) H n is K n -free, (ii) any finite K n -free graph is isomorphic to an induced subgraph of H n, (iii) any graph isomorphism between finite subsets of H n extends to an automorphism of H n. Given n 3, H n can also be constructed as the Fraïssé limit of the clas of finite K n -free graphs. WestudygraphstructuresinthegraphlanguageL = {R},whereRisinterpreted as the binary edge relation. We consider (1) T 0, the complete theory of the random graph, (2) T n = Th(H n ), for n 3. It is a well-known fact (and a standard exercise) that each of these theories is ℵ 0 -categorical with quantifier elimination. Fix n 3 and fix H n = T n, a sufficiently saturated monster model of T n. As H n is a graph, we can embed it in a larger sufficiently saturated monster model G = T 0. Note that H n isthen asubgraphofg, but not an elementarysubstructure. Let κ(h n ) = sup{κ : H n is κ-saturated}. For the rest of the paper, n, H n, and G are fixed. By saturation, we have the following fact. Proposition 3.2. Suppose H and X G, such that X is K n -free, X, and X κ(h). Then there is a graph embedding f : X H such that f = id. The remainder of this section is devoted to specifying notation and conventions concerning the language L. First, we consider types. Definition 3.3. Suppose G, with < κ(h n ). (1) We only consider partial types π( x) such that x κ(h n ). Furthermore, we will assume types are symmetricially closed. For example if c then xrc π( x) if and only if crx π( x). (2) An R-type over is a collection π( x) of atomic and negated atomic L- formulas, none of which is of the form x i = c, where c. When we say that π( x,ȳ) is an R-type over, we will assume further that π( x,ȳ) does not contain x i = x j or y i = y j, for some i j. (3) Suppose π( x) is an R-type over. An optimal solution of π( x) is a tuple ā = π( x) such that (i) a i a j for all i j and a i for all i, (ii) a i Ra j if and only if x i Rx j π( x), (iii) given c, a i Rc if and only if x i Rc π( x). We will frequently use the following fact, which says that we can always find optimal solutions of R-types. Proposition 3.4. Suppose H n and π( x) is an R-type over. (a) π( x) is consistent with T 0 if and only if it has an optimal solution in G. (b) π( x) is consistent with T n if and only if it has an optimal solution in H n.

4 4 GABRIEL ONANT This is a straightforward exercise, which we leave to the reader. The idea is that a type cannot prove than an edge exists in a graph, without explicitly saying so. Moreover, removing extra edges to optimize the solution of a consistent type is always possible and, in the case of T n, will not conflict with the requirement that the solution be K n -free. Next, we specify notation and conventions concerning L-formulas. Definition 3.5. Suppose G. (1) Let L 0 () be the collection of conjunctions of atomic and negated atomic L-formulas, with parameters from, such that no conjunct is of the form x = c, where x is a variable and c. When we write ϕ( x,ȳ) L 0 (), we will assume further that no conjunct of ϕ( x,ȳ) is of the form x i = x j or y i = y j, for some i j. (2) Given ϕ( x) L 0 () and θ( x), an atomic or negated atomic formula, we write ϕ( x) θ( x) if θ( x) is a conjunct of ϕ( x). (3) We will assume L 0 ()-formulas are symmetrically closed. For example ϕ( x) xrc if and only if ϕ( x) crx. (4) Let L R () be the collection of formulas ϕ( x,ȳ) L 0 () such that no conjunct is of the form x i = y j. The main result of this paper will be a characterizationofforking and dividing in T n. We will use the following characterizationof dividing in T 0, which is a standard exercise (see e.g. [9]). Fact 3.6. Fix G and ϕ( x,ȳ) L 0 (). Suppose b G\ is such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over if and only if ϕ( x, b) x i = b for some b b. onsequently, if A,B, G then A d B A B. T 0 is a standard example of a simple theory, and so the previous fact is also a characterization of forking. On the other hand, T n is non-simple. Indeed, the Henson graph is a canonical example where f fails amalgamation over models (see [6]). A direct proof of this (for n = 3) can be found in [4, Example 2.11(4)]. The precise classification of T n is well-known, and summarized by the following result. Fact 3.7. T n is TP 2, SOP 3, and NSOP 4. See [2] and [8] for definitions of these properties. The proof of TP 2 can be found in [2] for n = 3. SOP 3 and NSOP 4 are demonstrated in [8] for n = 3. The generalizations of these arguments to n 3 are fairly straightforward. However, NSOP 4 for all n 3 also follows from a more general result in [7]. 4. Dividing in T n The goal of this section is to find a graph theoretic characterization of dividing independence in T n. Therefore, when we say that a partial type divides over H n, we mean with respect to the theory T n. Lemma 4.1. Suppose ( b l ) l<ω is an indiscernible sequence in H n. (a) H n = b k i Rbl i for all k < l < ω and 1 i b 0. (b) If l( b 0 ) < n 1 then l<ω b l is K n 1 free.

5 FORKING AND DIVIDING IN HENSON GRAPHS 5 Proof. Part (a). Suppose not. By indiscernibility, H n = b k i Rbl i for all k < l < ω. In particular, {b 1 i,...,bn i } = K n, which is a contradiction. Part (b). Suppose S B := l<ω b l with S = n 1 and, for 1 i b 0, let S i = S {b l i : l < ω}. Since l( b 0 ) < n 1, there is some i such that S i 2. By part (a), S = Kn 1. We first define a graph theoretic binary relation on disjoint graphs, which will capture the notion of dividing in T n. Definition 4.2. (1) Suppose B, H n are disjoint. Then B is n-bound to, written K n (B/), if there is B 0 B such that (i) B 0 = n and B 0 B 0 B, (ii) if u,v B 0 are distinct then urv, (iii) if u B 0 B and v B 0 then urv. We say B 0 witnesses K n (B/). Informally, B 0 witnesses K n (B/) if and only if the only thing preventing B 0 = Kn is a possible lack of edges between vertices in B. (2) Suppose ϕ( x,ȳ) L R () and b H n \ such that ϕ( x, b) is consistent. Then b is ϕ-n-bound to, written K ϕ n ( b/), if there is B b, with 0 < B < n, such that (i) K n (B/), (ii) K n (B/ā) for all ā = ϕ( x, b). We say B witnesses K ϕ n( b/). The main resultofthis section(theorem 4.5)will showthat Kn ϕ is the graphtheoretic interpretation of dividing. In particular, for ϕ( x,ȳ) L R () and b H n \ with ϕ( x, b) consistent, we will show that ϕ( x, b) divides over if and only if Kn ϕ( b/). The reverse direction of the proof of this will use the following construction of a particular indiscernible sequence. onstruction 4.3. Fix H n and b H n \. We extend b to an infinite - indiscernible sequence ( b l ) l<ω, such that b l i bm j and b l i Rbm j for all l < m < ω and 1 i,j b. Note that ( b l ) l<ω is K n -free and so ( b l ) l<ω is an indiscernible sequence in H n. Given B b, we let Γ( b,b) be the graph expansion of ( b l ) l<ω obtained by adding b l i Rbm j if and only if l < m, i < j, and b i,b j B. We can embed Γ( b,b) into G over. Furthermore, if Γ( b,b) is K n -free, then we can embed Γ( b,b) in H n over. In this case, if Γ 0 ( b,b) is the image of ( b l ) l<ω, then Γ 0 ( b,b) is a -indiscernible sequence in H n. Lemma 4.4. Let H n and b H n \. Suppose ϕ( x,ȳ) L R () such that ϕ( x, b) is consistent and K ϕ n ( b/), witnessed by B b. (a) Γ( b,b) is K n -free. (b) If Γ 0 ( b,b) = ( b l ) l<ω then {ϕ( x, b l ) : l < ω} is (n 1)-inconsistent with T n. Proof. We may consider Γ 0 ( b,b) as an indiscernible sequence in G. Part (a). Suppose K n = W Γ( b,b). Since is Kn -free, W Γ 0 ( b,b). Say W Γ 0 ( b,b) = {b l1 i 1,...,b lr i r } with l 1... l r. Note that i s i t for all 1 s < t r by Lemma 4.1. Let B 0 = {b i1,...,b ir }. Define V = (W\{b l1 i 1,...,b lr i r }) {b i1,...,b ir }.

6 6 GABRIEL ONANT If l 1 = l r then, since b l1 b, it follows that V = Kn, which is a contradiction. Therefore l 1 < l r. By construction of Γ( b,b) it follows that b i1,b ir B. If 1 s r then we either have l 1 < l s or l s < l r, and in either case it follows that b is B. Therefore r B n 1; in particular W. But then V witnesses that B is n-bound to, which is a contradiction. Therefore Γ( b,b) is K n -free. Part (b). By part (a), we may consider Γ 0 ( b,b) as in indiscernible sequence in H n. By indiscernibility, it suffices to show that the R-type π( x) = {ϕ( x, b l ) : 0 l < n 1} is inconsistent with T n. So suppose, towards a contradiction, that π( x) is consistent with T n and let ā H n be an optimal solution. Then ā = ϕ( x, b) so, by assumption, there is D Bā witnessing K n (B/ā). We have D B. Moreover, K n (B/) implies D ā. To ease notation, let D B = {b 0,...,b k } and D ā = {a 0,...,a m }. Note that k < n 1. Define A 0 = D ā and B 0 = {b 0 0,...,bk k }. We make the following observations. (1) If u,v A 0 are distinct then urv. (2) If b i i,bj j B 0 are distinct, then b i i Rbj j by construction of Γ( b,b). (3) If c A 0 and b j j B 0 then b j j Rc since b jrc and b j b. (4) If a i A 0 ā and b j j B 0 then, since ā is an optimal solution of π( x), we have a i Rb j ( ) ) ϕ( x, b) x i Rb j (ϕ( x, b j ) x i Rb j j a i Rb j j. TheseobservationsimplyA 0 B 0 = Kn,whichisacontradictionsinceA 0 B 0 H. Theorem 4.5. Let H n, ϕ( x,ȳ) L R (), and b H n \ such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over if and only if K ϕ n( b/). Proof. ( ): Suppose B b witnesses K ϕ n( b/). Then Γ( b,b) H n and {ϕ( x, b l ) : l < ω} is (n 1)-inconsistent by Lemma 4.4. So ϕ( x, b) divides over. ( ): Suppose ϕ( x, b) divides over. Then there is a -indiscernible sequence ( b l ) l<ω, with b 0 = b, such that {ϕ( x, b l ) : l < ω} is inconsistent. Let G = l<ω b l. onsider G as a subgraph of G, and note that ( b l ) l<ω is still -indiscernible in G. Since ϕ( x,ȳ) L R () and ϕ( x, b) is consistent (in G), it follows from 3.6 that ϕ( x, b) does not divide over in G. Therefore there is an optimal realization d G of π( x) := {ϕ( x, b l ) : l < ω}. If G d is K n - free then G d embeds in H n over G, which is a contradiction. Therefore there is K n = W G d. Note that W d since G is Kn -free. Without loss of generality, let W d = {d 1,...,d m }. Suppose, towards a contradiction, that W G. Let ā H n be a solution to ϕ( x, b). Since d is an optimal realization of π( x), we make the following observations. (1) If 1 i,j m are distinct then a i Ra j. (2) If b W G then a i Rb for all 1 i m. Therefore, K n = (W G) {a1,...,a m }, which is a contradiction, and so W (G\).

7 FORKING AND DIVIDING IN HENSON GRAPHS 7 Let W (G\) = {b l1 j 1,...,b l k jk } and note that 1 t < n. By Lemma 4.1, j s j t for all s t, so without loss of generality, let W (G\) = {b l1 1,...,bl k k }. Let B = {b 1,...,b k }. laim 1: K n (B/). Proof: Suppose X B witnesses K n (B/). By indiscernibility, if B 0 = {b ls s : b s B X}, then (X ) B 0 witnesses that B 0 is n-bound to. But b ls s Rb lt t for all s t, so (X ) B 0 = Kn, which is a contradiction. laim 2: If ā H n is a solution of ϕ( x, b) then K n (B/ā). Proof: We show (W ) B {a 1,...,a m } witnesses K n (B/ā), which means verifying all of the necessary relations. Recall that d is an optimal solution to π( x). (1) If 1 i j m then d i Rd j ( ) ϕ( x, b) x i Rx j ai Ra j. (2) If 1 i m and c W then d i Rc ( ϕ( x, b) x i Rc ) a i Rc. (3) If 1 i m and 1 s k then d i Rb ls s ( ) ( ) ϕ( x, b ls ) x i Rb ls s ϕ( x, b) xi Rb s ai Rb s. Together, laims 1 and 2 imply K ϕ n ( b/), as desired. We can now give the full characterization of nondividing formulas in T n, and the ternary relation d on sets, which gives the analogy of Fact 3.6 for T n. Theorem 4.6. (a) Suppose H n, ϕ( x,ȳ) L 0 (), and b H n \ such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over if and only if ϕ( x, b) x i = b for some b b, or ϕ( x,ȳ) L R () and Kn( b/). ϕ (b) Suppose A,B, H n. Then A d B if and only if (i) A B, and (ii) for all b B\, if K n ( b/a) then K n ( b/). Proof. Part(a) follows immediately from Theorem 4.5. For part(b), let ā enumerate A. Note that ā is an optimal solution to tp(ā/b). ( ): Suppose A d B. Then A B is immediate. For condition (ii), fix b B\ and suppose, towards a contradiction, that Kn ( b/) and K n ( b/a). Fix X A b witnessing K n ( b/a). Let 0 = X. Without loss of generality, let X A = {a 1,...,a m } and X b = b = {b 1,...,b k }. Define L R ()-formula ϕ( x,ȳ), where x i = (x 1,...,x m ), ȳ = (y 1,...,y k ), and ϕ( x,ȳ) expresses x is a complete graph and xrȳ, xr 0 and ȳr 0, 0 is a complete graph. Then Kn ϕ( b /) and ϕ( x, b ) tp(a/b). Therefore A d B by Theorem 4.5, which is a contradiction. ( ): Suppose A d B. Then there is some ϕ( x,ȳ) L 0() and b B\ such that ϕ( x, b) divides over and ϕ( x, b) tp(a/b). If ϕ( x,ȳ) x i = y j for some i,j, then a i = b j (A B)\ and (i) fails. Otherwise, ϕ( x,ȳ) L R () and Kn( b/). ϕ It follows that K n ( b/) and K n ( b/a), so (ii) fails. The theorem translates the model theoretic notion of dividing to the graph theoretic notion K ϕ n( b/). Although the definition of K ϕ n( b/) implies that we must check all solutions of ϕ, it suffices to check an optimal one.

8 8 GABRIEL ONANT orollary 4.7. Let H n, ϕ( x,ȳ) L R (), and b H n \ such that ϕ( x, b) is consistent. Let ā be an optimal solution. Then ϕ( x, b) divides over if and only if there is B b such that K n (B/) and K n (B/ā). Proof. By Theorem 4.5, we need to show K ϕ n( b/) if and only if there is B b such that K n (B/) and K n (B/ā). The forward direction is clear. onversely, suppose B b such that K n (B/) and K n (B/ā). Let d be any solution to ϕ( x, b). We want to show K n (B/ d). Let B 0 Bā witness K n (B/ā). Define 0 = (B 0 ) and D = {d i : a i B 0 ā}. Since ā is optimal, we can make the following observations to show that B 0 0 D witnesses K n (B/ d). (1) If c 1,c 2 0 then c 1 Rc 2 by assumption. (2) If c 0 and d i D then a i Rc ( ϕ( x, b) x i Rc ) d i Rc. (3) If c 0 and b B 0 then brc by assumption. (4) If d i D and b B 0 then a i Rb ( ϕ( x, b) x i Rb ) d i Rb. We end this section by giving some examples and traits of dividing formulas in T n. We will begin using the following notation. If A and B are sets we write ARB to mean arb for all a A and b B. On other hand, A RB means arb for all a A and b B. orollary 4.8. Suppose H n and b 1,...,b n 1 H n \ are distinct. Then the formula ϕ(x, b) := divides over if and only if K n ( b/). n 1 i=1 xrb i Proof. First, if b = K n 1 then ϕ(x, b) is inconsistent and thus divides over. Furthermore, in this case K n ( b/) since, if so, then there is some c such that cr b and so c b = K n. Therefore we may assume b = Kn 1. ( ): If ϕ(x, b) divides over then, by Theorem 4.5, there is some B b such that K n (B/) and K n (B/a) for any a = ϕ(x, b). Let a = ϕ(x, b) such that a R. Let X Ba witness K n (B/a). Then K n (B/) implies a X, and so X = since a R. Therefore X Ba ba, X = n, and b = n 1. It follows that B = b, and so K n ( b/). ( ): Note that if a realizes ϕ(x, b), then K n ( b/a). So if K n ( b/) then b itself witnesses Kn( b/). ϕ By Theorem 4.5, ϕ(x, b) divides over. orollary 4.9. Let H and ϕ( x,ȳ) L R () such that ϕ( x,ȳ) x i Ry j for all i,j. If b H\ such that ϕ( x, b) is consistent, then ϕ( x, b) does not divide over. Proof. Suppose B b is such that K n (B/). Let ā be an optimal solution to ϕ( x, b). Then ā RB, so K n (B/ā). By orollary 4.7, ϕ( x, b) does not divide over. orollary Let H and ϕ( x,ȳ) L R (). Suppose b H\ such that ϕ( x, b) is consistent and divides over. Define R ϕ = {b b : ϕ( x, b) x i Rb for some i} {x i : ϕ( x, b) x i Rb for some b b}. Then R ϕ n and b R ϕ > 1.

9 FORKING AND DIVIDING IN HENSON GRAPHS 9 Proof. By assumption, we have Kn ϕ( b/). If ā is an optimal solution of ϕ( x, b) then there is some X bā witnessing K n ( b/ā). Note that X ā since K n ( b/). Set B = (X b) {x i : a i X}. Then B n and B R ϕ since ā is optimal. Finally, b B = b X > 1, since otherwise X = K n. orollary4.10saysthat ifaformulafroml R () dividesthen it needs to mention edges between at least n vertices (and more than one parameter). This is not surprising since no consistent formula from L R () will divide in T 0, and so dividing in T n should come from the creation of a graph that is too close to K n. 5. Forking for complete types In this section, we use our characterization of d in T n to show that forking and dividing are the same for complete types. The proof takes two steps, the first of which is to prove full existence for the following ternary relation on graphs. We take the following definition from [1]. Definition 5.1. Given A,B, H n, define edge independence by A R B A B and there is no edge from A\ to B\. Lemma 5.2. For all A,B, H n there is A A such that A R B. Proof. Fix A,B, H n and enumerate A\(B) = (a i ) i<λ. We define a graph G = B(a i ) i<λ G, where each a i is a new vertex, and (i) for all i,j < λ, a i Ra j if and only if a ira j, (ii) for all i < λ and c, a i Rc if and only if a irc, (iii) for all i < λ and b B\, a i Rb. We claim that G is K n -free. Indeed, if K n = W G then, by (iii), it follows that W B or W (a i ) i<λ. But B H n so this means W (a i ) i<λ, which, by (i) and (ii), means A is not K n -free, a contradiction. Therefore G embeds in H n over B. Let (a ) i<λ be the image of (a i ) i<λ and set A = (A ) (a i ) i<λ. Then it is clear that A A and A R B. Using full existence of R, we can prove the full characterization of forking and dividing in T n. Theorem 5.3. Suppose A,B, H n. Then A f B A d B A B and, for all b B\, K n ( b/a) implies K n ( b/). Proof. The second equivalence is by Theorem 4.6; and dividing implies forking in any theory. Therefore we only need to show A d B implies A f B. Suppose A f B. Then there is some D H n\b such that A d BD for any A B A. By Lemma 5.2, let A B A such that A R D. By assumption, we have B A d BD. ase 1: A BD. We have A BD B by assumption, so this means there is b (A B)\. But A B A and so b (A B)\. Therefore A d B, as desired. ase 2: A BD. Then, since A d BD, it follows from Theorem 4.6 that there is b BD\ such that K n ( b/) and K n ( b/a ). Let X A b witness

10 10 GABRIEL ONANT K n ( b/a ). Note that X A BD. Moreover, note also that if X (A \B), then X BD, and so X witnesses K n ( b/), which is a contradiction. Therefore X (A \B). Then we claim that X A B. Indeed, otherwise there is u X (A \B) and v X (D\A B). Therefore u v, u A, and v b, and so urv, since X witnesses K n ( b/a ). But this contradicts that there is no edge from A \B to D\B. So we have X A B. Let b = X b B\. Then K n ( b/) implies K n ( b /), and X witnesses K n ( b /A). Therefore A d B. Since A B A, we have A d B, as desired. It is a generalfact that if d = f in some theory T, then all sets are extension basesfornonforking. Indeed, ifapartialtypeforksover thenitcanbeextendedto a complete type that forks(and therefore divides over ). Therefore, by Proposition 2.2(b), no partial type forks over its own set of parameters. 6. A forking and nondividing formula in T n We have shown that forking and dividing are the same for complete types in T n. In this section, we show that the same result cannot be obtained for partial types, by demonstrating an example of a formula in T n that forks, but does not divide. Lemma 6.1. Suppose ( b l ) l<ω is an indiscernible sequence in G such that l( b 0 ) = 4 and b 0 is K 2 -free (no edges). Then either there are i < j such that {b l i,bl j : l < ω} is K 2 -free, or l<ω b l is not K 3 -free. Proof. Let B = l<ω b l. Suppose first that for all i < j, we have b 0 i Rb1 j or b1 i Rb0 j. Let f : {(i,j) : 1 i < j 4} {0,1} such that f(i,j) = 0 b 0 i Rb1 j. laim: If B is K 3 -free then for all i < j < k, f(i,j) = f(j,k). Proof: Suppose not. ase 1: f(i,j) = 1 and f(j,k) = 0. If f(i,k) = 0 then by indiscernibility we have b 1 i Rb0 j, b0 j Rb2 k and b1 i Rb2 k ; and so {b1 i,b0 j,b2 k } = K 3. If f(i,k) = 1 then by indiscernibility we have b 2 i Rb0 j, b0 j Rb1 k and b2 i Rb1 k ; and so {b2 i,b0 j,b1 k } = K 3. ase 2: f(i,j) = 0andf(j,k) = 1. Iff(i,k) = 0then{b 0 i,b2 j,b1 k } = K 3. Otherwise, if f(i,k) = 1 then {b 1 i,b2 j,b0 k } = K 3. By the claim, if f(1,2) = 0 then f(2,3), f(3,4), f(1,3) and f(2,4) are all 0; and so {b 0 1,b1 2,b2 3 } = K 3. If f(1,2) = 1 then f(2,3), f(3,4), f(1,3), and f(2,4) are all 1; and so {b 2 1,b 1 2,b 0 3} = K 3. In any case we have shown that B is not K 3 -free. So we may assume that there is some i < j such that b 0 i Rb1 j and b1 i Rb0 j. By indiscernibility, and since b 0 i Rb0 j, it follows that {bl i,bl j : l < ω} is K 2-free. Theorem 6.2. Let, b = (b 1,b 2,b 3,b 4 ) H n such that = K n 3, br, and b is K 2 -free. For i < j, let ϕ i,j (x,b j,b j ) = xrb i b j. Set ϕ(x, b) = i<j ϕ i,j(x,b i,b j ). Then ϕ(x, b) forks over but does not divide over. Proof. For any i j, b i b j = n 1, and so K n (b i,b j /). Moreover, we clearly have that if a = ϕ(x,b i,b j ) then K n (b i,b j /a). By Theorem 4.5, ϕ i,j (x,b i,b j ) divides over, and therefore ϕ(x, b) forks over. Let ( b l ) l<ω be -indiscernible, with b 0 = b. If there is some K 3 = W l<ω b l then K n = W, since bl b for all l < ω implies RW. Therefore l<ω b l is

11 FORKING AND DIVIDING IN HENSON GRAPHS 11 K 3 -free and so, by Lemma 6.1, there are i < j such that B := {b l i,bl j : l < ω} is K 2- free. Since = n 3, it follows that B is K n 1 -free and so there is some a H such that ar(b). But then a = {ϕ i,j (x,b l i,bl j ) : l < ω} {ϕ(x, b l ) : l < ω}. By Theorem 2.2, ϕ(x, b) does not divide over A. 7. Final remarks We have shown that T n is an NSOP 4 theory in which all sets are extension bases for nonforking, but forking and dividing are not always the same. This only partially answers the question of how the results of [3] extend to theories with TP 2. Inpartictular, forkingis the sameasdividingforcomplete types int n, which means there is good behavior of nonforking beyond just the fact that all sets are extension bases. This leads to the following amended version of the main question. Question 7.1. Suppose that in some theory all sets are extension bases for nonforking. (1) Does NSOP 3 imply forking and dividing are the same for partial types? (2) For what classes of theories do we have f = d? References [1] Hans Adler, A geometric introduction to forking and thorn-forking, J. Math. Log. 9 (2009), no. 1, MR (2011i:03026) [2] Artem hernikov, Theories without the tree property of the second kind, arxiv: v3 [math.lo], [3] Artem hernikov and Itay Kaplan, Forking and dividing in NTP 2 theories, J. Symbolic Logic 77 (2012), no. 1, MR [4] Bradd Hart, Stability theory and its variants, Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, ambridge Univ. Press, ambridge, 2000, pp MR (2001h:03058) [5]. Ward Henson, A family of countable homogeneous graphs, Pacific J. Math. 38 (1971), MR (46 #3377) [6] Byunghan Kim and Anand Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), no. 2-3, , Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR (99b:03049) [7] Rehana Patel, A family of countably universal graphs without SOP 4, preprint, [8] Saharon Shelah, Toward classifying unstable theories, Ann. Pure Appl. Logic 80 (1996), no. 3, MR (97e:03052) [9] Katrin Tent and Martin Ziegler, A course in model theory, Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, A, MR Department of Mathematics, Statistics and omputer Science, University of Illinois at hicago, hicago, IL, 60607, USA address: gconan2@uic.edu