Forking and Dividing in Random Graphs

Transcription

1 Forking and Dividing in Random Graphs Gabriel Conant UIC Graduate Student Conference in Logic University of Notre Dame April 28-29, 2012 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

2 Definitions (a) A formula ϕ( x, b) divides over a set A M if there is a sequence ( b l ) l<ω and k Z + such that b l A b for all i < ω and {ϕ( x, b l ) : l < ω} is k-inconsistent. (b) A partial type divides over A if it proves a formula that divides over A. (c) A partial type forks over A if it proves a finite disjunction of formulas that divide over A. (d) Given a theory T and M = T, we define the ternary relations d and on { c M} {A M} 2 by c A tp( c/ab) does not divide over A, c A tp( c/ab) does not fork over A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

3 Facts Theorem A partial type π( x, b) (with π( x, ȳ) a type over A) divides over A if and only if there is a sequence ( b l ) l<ω of indiscernibles over A with b 0 = b and l<ω π( x, b l ) inconsistent. Definition A theory T is simple if there is some formula ϕ( x, ȳ) with the tree property. Theorem A theory T is simple if and only if for all B M and p S n (B) there is some A B with A T such that p does not divide over A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

4 Setting Let L = {R}, where R is a binary relation symbol. We write xry, rather than R(x, y). If A and B are sets, we write ARB to mean that every point in A is connected to every point in B; and A RB to mean that no point in A is connected to any point in B. We write A M to mean that A is a subset of M with A < M. Given A M, let L 0 A L A be the set of conjunctions of atomic and negated atomic formulas such that no conjunct is of the form x i = a, for some variable x i and a A. When we refer to formulas ϕ( x, ȳ) in L 0 A, we will assume no conjunct is of the form x i = x j or y i = y j for distinct i, j. Let L R A be the set of L0 A-formulas ϕ( x, ȳ) containing no conjunct of the form x i = y j for any i, j. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

5 The Random Graph We let T 0 be the theory of the random graph, i.e., T 0 contains 1 the axioms for nonempty graphs, x xrx x y(xry yrx) x x = x; 2 extension axioms asserting that if A and B are finite disjoint sets of vertices then there is some c such that cra and c RB. T 0 is a complete, ℵ 0 -categorical theory with quantifier elimination and no finite models. We fix an uncountable saturated monster model M = T 0 and define the ternary relation on { c M} {A M} 2 such that c A B c B A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

6 The Random Graph is Simple (first proof) Theorem (Kim & Pillay) A theory T is simple if and only if there is a ternary relation o satisfying automorphism invariance, extension, local and finite character, symmetry, transitivity, monotonicity, and independence over models. Moreover, in this case o, d, and are all the same. Independence over models: Let M = T, M A, B and A o B. M Suppose ā, b M such that ā M b, ā o A, and b M o B. Then there M is c M such that tp(ā/a) tp( b/b) tp( c/ab) and c o AB. M Proposition The ternary relation satisfies all the necessary properties to witness that T 0 is simple. In particular, we have c A B c B A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

7 The Random Graph is Simple (second proof) Lemma Let A M, b, c M such that c b. If ( b l ) A l<ω is A-indiscernible, with b 0 = b, then there is d M with d b A l<ω l such that d b l A c b for all l < ω. Theorem Suppose B M = T 0 and p S n (B). Then p does not divide over A = {b B : i x i = b p( x)}. 1 A theory T is supersimple if for all p S n (B) there is some finite A B such that p does not divide over A. Therefore T 0 is supersimple. 2 (Kim) If is replaced by in the above lemma, the statement remains true in any simple theory. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

8 Dividing in the Random Graph Theorem Let A M = T 0, ϕ( x, ȳ) L 0 A, and b M such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over A if and only if ϕ( x, b) x j = b for some b b\a. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

9 The K n -free Random Graph For n 3, we let T n be the theory of the K n -free random graph, i.e., T n contains 1 the axioms for nonempty graphs; 2 a sentence asserting that the graph is K n -free; 3 extension axioms asserting that if A and B are finite disjoint sets of vertices, and A is K n 1 -free, then there is some c such that cra and c RB. T n is a complete, ℵ 0 -categorical theory with quantifier elimination and no finite models. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

10 The K n -free Random Graph is Not Simple (first proof) Theorem d -independence over models fails in T n. Proof. b 1 b 2 b 3 b n 1... Claim: If b M\A, b RA, l( b) < n 1, and p is in S 1 (A b), then p does not divide over A. a 1 a 2 a 3 a n 1 M So a i d M {a j : j < i} and b i d M a i. Clearly, {tp(b i /Ma i ) : i < n} cannot be amalgamated. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

11 The K n -free Random Graph is Not Simple (second proof) Definition Let k 3. A theory T has the k-strong order property, SOP k, if there is a formula ϕ( x, ȳ), with l( x) = l(ȳ), and ( b l ) l<ω such that M = ϕ( b l, b m ) l < m < ω; M = x 1... x k (ϕ( x 1, x 2 )... ϕ( x k 1, x k ) ϕ( x k, x 1 )). Fact:... SOP k+1 SOP k... SOP 3 tree property. Theorem For all n 3, T n has SOP 3. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

12 The K n -free Random Graph is Not Simple (second proof) Proof (T n has SOP 3 ). K m = x 1 ȳ 1 = K m Assume n 4. Let m = n 3. Then 2m < n. Let ϕ( x 1, x 2, x 3, ȳ 1, ȳ 2, ȳ 3 ) describe this configuration. K m = x 2 ȳ 2 = K m We can construct an infinite chain in M = T n. No 3-cycle is possible. K m = x 3 ȳ 3 = K m Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

13 The K n -free Random Graph is Not Simple (second proof) Definition Let k 3. A theory T has the k-strong order property, SOP k, if there is a formula ϕ( x, ȳ) and ( b l ) l<ω such that M = ϕ( b l, b m ) l < m < ω; M = x 1... x k (ϕ( x 1, x 2 )... ϕ( x k 1, x k ) ϕ( x k, x 1 )). Fact:... SOP k+1 SOP k... SOP 3 tree property. Theorem For all n 3, T n has SOP 3. Fact: For all n 3, T n does not have SOP 4. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

14 Recall: Dividing in the Random Graph Theorem Let A M = T 0, ϕ( x, ȳ) L 0 A, and b M such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over A if and only if ϕ( x, b) x j = b for some b b\a. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

15 Dividing in T n Definition Suppose A, B M are disjoint. Then B is n-bound to A if there is B 0 A B, B 0 = n, B 0 A B 0 B, such that 1 (B 0 A)R(B 0 B), 2 (B 0 A) = K m, where m = B 0 A. Informally, B is n-bound to A if there is a subgraph B 0 AB of size n, such that the only thing preventing B 0 from being isomorphic to K n is a possible lack of edges between points in B. A A b is 4-bound to A b is not 4-bound to A (but A is 4-bound to b) b 1 b 2 b 1 b 2 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

16 Dividing in T n Theorem Let A M, ϕ( x, ȳ) L R A, and b M\A such that ϕ( x, b) is consistent. Then ϕ( x, b) divides over A if and only if 1 b is not n-bound to A, 2 b is n-bound to A c for any c = ϕ( x, b). Proof. ( ) : b not n-bound to A allows construction of a sequence ( b l ) l<ω, with enough edges, so that being n-bound to A c will force (n 1)-inconsistency in {ϕ( x, b l ) : l < ω}. ( ): Let ( b l ) l<ω, indiscernible over A, witness dividing. Since ϕ( x, b) does not divide over A in T 0, there must be a copy of K n in A c( b l ) l<ω, where c is an optimal solution of {ϕ( x, b l ) : l < ω} in T 0. By indiscernibility, this K n will project to the required conditions in A b. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

17 Dividing in T n - Examples Corollary Suppose A M and b 1,..., b n 1 M\A are distinct. Then the formula ϕ(x, b) := n 1 i=1 xrb i divides over A if and only if b is not n-bound to A. Corollary Let A M and ϕ( x, ȳ) L R A. Suppose b M\A such that ϕ( x, b) is consistent and divides over A. Define R ϕ = {u A b : i ϕ( x, b) x i Ru} {x i : u A b, ϕ( x, b) x i Ru}. Then R ϕ n and b R ϕ > 1. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

18 Recall: in T 0 Theorem Let A, B, c M = T 0. Then c A B c d A B c B A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

19 in T n Lemma Let A M = T n and p S m (A). If p x = b for some b M then b A. If p xrb for some b M, then either b A or p x = a for some a A. Theorem Let A, B, c M = T n. Then c A B c d A B c B A and for all b B\A, b is either n-bound to A or not n-bound to A c. Corollary Let A M = T n. If π( x) is a partial type over A, then π( x) does not fork over A. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

20 Forking in T n Lemma Suppose ( b l ) l<ω is an indiscernible sequence in M = T n such that l( b 0 ) = 4 and b 0 is K 2 -free. 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. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

21 Forking in T n Theorem Let M = T n and A b M such that l( b) = 4, bra, A = K n 3, and b is K 2 -free. For i j, let ϕ i,j (x, b i, b j ) = xrb i xrb j a A xra, and set ϕ(x, b) := i j ϕ i,j (x, b i, b j ). Then ϕ(x, b) forks over A but does not divide over A. A = K n 3 ϕ(x, b) = xra {b i : xrb i } = 2 b 1 b 2 b 3 b 4 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

22 Higher Arity Graphs Now suppose R is a relation of arity r 2. An r-graph is an L-structure satisfying the following sentence x 1... x r R(x 1,..., x r ) x i x j R(x σ(1),..., x σ(r) ). i j σ S r So R can be thought of as a collection of r-element subsets of the r-graph. Let T r 0 and T r n, for n > r, be the natural analogs of T 0 and T n, respectively, to r-graphs. Call T r 0 the theory of the random r-graph, and T r n the theory of the random K r n-free r-graph, where K r n is the complete r-graph of size n. T r n can be thought of as the theory of the unique (countable) Fraïssé limit of the class of finite K r n-free r-graphs. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

23 Recall - Dividing in T 0 Lemma Let A M = T 0, b, c M such that c A b. If ( b l ) l<ω is A-indiscernible, with b 0 = b, then there is d M with d such that d b l A c b for all l < ω. A l<ω b l Lemma Let A M = T 0 and p( x, ȳ) S(A) such that x i y j p( x, ȳ) for all i, j. Suppose p( x, b) is consistent and ( b l ) l<ω is indiscernible over A with b 0 = b. Then there is a solution c of l<ω p( x, b l ). Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

24 Dividing in T r 0 Lemma Let A M = T r 0 and p( x, ȳ) S(A) such that x i y j p( x, ȳ) for all i, j. Suppose p( x, b) is consistent and ( b l ) l<ω is indiscernible over A with b 0 = b. Then there is a solution c of l<ω p( x, b l ). In fact, we can take c to be an optimal solution, in particular if A 0 A( b l ) l<ω, then for any i 1,..., i k M = R(c i1,..., c ik, A 0 ) R(x i1,..., x ik, A 0 ) p( x, b l ). l<ω Corollary (T0 r is simple.) Suppose B M = T0 r and p S(B). Then p does not fork over A = {b B : i x i = b p( x)}. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

25 Dividing in T r n, r > 2 Lemma Assume r > 2. Let A M = T r n and p( x, ȳ) S(A) such that x i y j p( x, ȳ) for all i, j. Suppose p( x, b) is consistent and ( b l ) l<ω is indiscernible over A with b 0 = b. Then there is a solution c of l<ω p( x, b l ). Corollary (T r n is simple if r > 2.) Suppose r > 2 and B M = T r n and p S(B). Then p does not fork over A = {b B : i x i = b p( x)}. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

26 b 4. Dividing in T r n, r > 2 Proof of Lemma. A b 0 b 1 b 2 b 3 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

27 Dividing in T r n, r > 2 Proof of Lemma. A c M = T0 r is an optimal solution to l<ω p( x, b l ). b 0 b 1 b 2 b 3 b 4. c Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

28 Dividing in T r n, r > 2 Proof of Lemma. b 0 A c M = T0 r is an optimal solution to l<ω p( x, b l ). Show A c( b l ) l<ω is K r n-free. b 1 b 2 b 3 b 4. c Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

29 Dividing in T r n, r > 2 Proof of Lemma. A b 0 b 1 b 2 W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. b 3 b 4. c Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

30 Dividing in T r n, r > 2 Proof of Lemma. A W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). b 0 b 1 b 2 b 3 b 4. c i c Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. There is c i W c. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

31 Dividing in T r n, r > 2 Proof of Lemma. A W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). b 0 b 1 b 2 b 3 b 4. u v c i c Show A c( b l ) l<ω is Kn-free. r Let Kn r = W A c( b l ) l<ω. There is c i W c. If l : W b l } > 1 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

32 Dividing in T r n, r > 2 Proof of Lemma. A W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). W 0 b 0 b 1 b 2 b 3 b 4. u v c i c Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. There is c i W c. If l : W b l } > 1 pick W 0 W \{c i, u, v}, with W 0 = r 3. Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

33 Dividing in T r n, r > 2 Proof of Lemma. A W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). W 0 b 0 b 1 b 2 b 3 b 4. u v c i c Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. There is c i W c. If l : W b l } > 1 pick W 0 W \{c i, u, v}, with W 0 = r 3. R(c i, u, v, W 0 ) l<ω p( x, b l ). Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

34 Dividing in T r n, r > 2 Proof of Lemma. A b m b 1 W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. b 2 b 3 b 4. c {l : W b l } 1 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

35 Dividing in T r n, r > 2 Proof of Lemma. A b m b 1 W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. b 2 b 3 b 4. c {l : W b l } 1 W = K r n p( c, b m ) Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

36 Dividing in T r n, r > 2 Proof of Lemma. A b m b 1 W = K r n c M = T0 r is an optimal solution to l<ω p( x, b l ). Show A c( b l ) l<ω is K r n-free. Let K r n = W A c( b l ) l<ω. b 2 b 3 c {l : W b l } 1 W = K r n p( c, b m ) b 4. This contradicts M = xp( x, b m ). Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27

37 References B. Kim & A. Pillay, Simple theories, Annals of Pure and Applied Logic 88 (1997) D. Marker, Model Theory: An Introduction, Springer, K. Tent & M. Ziegler, A Course in Model Theory, Cambridge, Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, / 27