Forking and Dividing in Random Graphs
Gabriel Conant
UIC
Graduate Student Conference in LogicUniversity of Notre Dame
April 28-29, 2012
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 1 / 27
Definitions
(a) A formula ϕ(x , b) divides over a set A ⊂M if there is a sequence(bl)l<ω and k ∈ Z+ such that bl ≡A b for all i < ω and{ϕ(x , bl) : l < ω} is k -inconsistent.
(b) A partial type divides over A if it proves a formula that divides overA.
(c) A partial type forks over A if it proves a finite disjunction offormulas 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 | dA
B ⇔ tp(c/AB) does not divide over A,
c |A
B ⇔ tp(c/AB) does not fork over A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 2 / 27
Facts
Theorem
A partial type π(x , b) (with π(x , y) a type over A) divides over A if andonly if there is a sequence (bl)l<ω of indiscernibles over A with b0 = band
⋃l<ω π(x , bl) inconsistent.
DefinitionA theory T is simple if there is some formula ϕ(x , y) with the treeproperty.
Theorem
A theory T is simple if and only if for all B ⊂M and p ∈ Sn(B) there issome A ⊆ B with |A| ≤ |T | such that p does not divide over A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 3 / 27
SettingLet 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 thatevery point in A is connected to every point in B; and A 6RB to meanthat 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 L0A ⊆ LA be the set of conjunctions of atomic and
negated atomic formulas such that no conjunct is of the form xi = a, forsome variable xi and a ∈ A. When we refer to formulas ϕ(x , y) in L0
A,we will assume no conjunct is of the form xi = xj or yi = yj for distincti , j .
Let LRA be the set of L0
A-formulas ϕ(x , y) containing no conjunct ofthe form xi = yj for any i , j .
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 4 / 27
The Random GraphWe let T0 be the theory of the random graph, i.e., T0 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 setsof vertices then there is some c such that cRA and c 6RB.
T0 is a complete, ℵ0-categorical theory with quantifier eliminationand no finite models.
We fix an uncountable saturated monster model M |= T0 and definethe 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, 2012 5 / 27
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 finitecharacter, symmetry, transitivity, monotonicity, and independence overmodels. Moreover, in this case | o, | d , and | are all the same.
Independence over models: Let M |= T , M ⊆ A,B and A | oM
B.Suppose a, b ∈M such that a ≡M b, a | o
MA, and b | o
MB. Then there
is c ∈M such that tp(a/A) ∪ tp(b/B) ⊆ tp(c/AB) and c | oM
AB.
PropositionThe ternary relation | ∩ satisfies all the necessary properties towitness that T0 is simple. In particular, we have
c |A
B ⇔ c ∩ B ⊆ A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 6 / 27
The Random Graph is Simple (second proof)
Lemma
Let A ⊂M, b, c ∈M such that c | ∩A
b. If (bl)l<ω is A-indiscernible,with b0 = b, then there is d ∈M with d | ∩
A
⋃l<ω bl such that
d bl ≡A cb for all l < ω.
Theorem
Suppose B ⊂M |= T0 and p ∈ Sn(B). Then p does not divide overA = {b ∈ B : ∃i xi = b ∈ p(x)}.
1 A theory T is supersimple if for all p ∈ Sn(B) there is some finiteA ⊆ B such that p does not divide over A. Therefore T0 issupersimple.
2 (Kim) If | ∩ is replaced by | in the above lemma, the statementremains true in any simple theory.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 7 / 27
Dividing in the Random Graph
Theorem
Let A ⊂M |= T0, ϕ(x , y) ∈ L0A, and b ∈M such that ϕ(x , b) is
consistent. Then ϕ(x , b) divides over A if and only if ϕ(x , b) B xj = bfor some b ∈ b\A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 8 / 27
The Kn-free Random GraphFor n ≥ 3, we let Tn be the theory of the Kn-free random graph, i.e.,
Tn contains
1 the axioms for nonempty graphs;2 a sentence asserting that the graph is Kn-free;3 “extension axioms” asserting that if A and B are finite disjoint sets
of vertices, and A is Kn−1-free, then there is some c such that cRAand c 6RB.
Tn is a complete, ℵ0-categorical theory with quantifier eliminationand no finite models.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 9 / 27
The Kn-free Random Graph is Not Simple (first proof)
Theorem
| d -independence over models fails in Tn.
Proof.b1 b2 b3 bn−1
a1 a2 a3 an−1
M
. . .Claim:If b ∈M\A, b 6RA, l(b) < n − 1, and p isin S1(Ab), then p does not divide over A.
So ai | dM{aj : j < i} and bi | d
Mai .
Clearly, {tp(bi/Mai) : i < n}cannot be amalgamated.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 10 / 27
The Kn-free Random Graph is Not Simple (second proof)
DefinitionLet k ≥ 3. A theory T has the k -strong order property, SOPk , if thereis a formula ϕ(x , y), with l(x) = l(y), and (bl)l<ω such that
M |= ϕ(bl , bm) ∀ l < m < ω;
M |= ¬∃x1 . . . ∃xk (ϕ(x1, x2) ∧ . . . ∧ ϕ(xk−1, xk ) ∧ ϕ(xk , x1)).
Fact: . . .⇒ SOPk+1 ⇒ SOPk ⇒ . . .⇒ SOP3 ⇒ tree property.
TheoremFor all n ≥ 3, Tn has SOP3.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 11 / 27
The Kn-free Random Graph is Not Simple (second proof)
Proof (Tn has SOP3).
x1
x2
x3
Km ∼=
Km ∼=
Km ∼=
y1
y2
y3
∼= Km
∼= Km
∼= Km
Assume n 6= 4.Let m = dn
3e. Then 2m < n.
Let ϕ(x1, x2, x3, y1, y2, y3)describe this configuration.
We can construct an infinitechain in M |= Tn.
No 3-cycle is possible.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 12 / 27
The Kn-free Random Graph is Not Simple (second proof)
DefinitionLet k ≥ 3. A theory T has the k -strong order property, SOPk , if thereis a formula ϕ(x , y) and (bl)l<ω such that
M |= ϕ(bl , bm) ∀ l < m < ω;
M |= ¬∃x1 . . . ∃xk (ϕ(x1, x2) ∧ . . . ∧ ϕ(xk−1, xk ) ∧ ϕ(xk , x1)).
Fact: . . .⇒ SOPk+1 ⇒ SOPk ⇒ . . .⇒ SOP3 ⇒ tree property.
TheoremFor all n ≥ 3, Tn has SOP3.
Fact: For all n ≥ 3, Tn does not have SOP4.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 13 / 27
Recall: Dividing in the Random Graph
Theorem
Let A ⊂M |= T0, ϕ(x , y) ∈ L0A, and b ∈M such that ϕ(x , b) is
consistent. Then ϕ(x , b) divides over A if and only if ϕ(x , b) B xj = bfor some b ∈ b\A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 14 / 27
Dividing in Tn
DefinitionSuppose A,B ⊂M are disjoint. Then B is n-bound to A if there is
B0 ⊆ A ∪ B, |B0| = n, B0 ∩ A 6= ∅ 6= B0 ∩ B, such that
1 (B0 ∩ A)R(B0 ∩ B),2 (B0 ∩ A) ∼= Km, where m = |B0 ∩ A|.
Informally, B is n-bound to A if there is a subgraph B0 ⊆ AB of size n,such that the only thing preventing B0 from being isomorphic to Kn is apossible lack of edges between points in B.
A A
b1 b2 b1 b2
b is 4-bound to Ab is not 4-bound to A(but A is 4-bound to b)
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 15 / 27
Dividing in Tn
Theorem
Let A ⊂M, ϕ(x , y) ∈ LRA , 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 Ac for any c |= ϕ(x , b).
Proof.(⇐) : b not n-bound to A allows construction of a sequence (bl)l<ω,with enough edges, so that being n-bound to Ac will force(n − 1)-inconsistency in {ϕ(x , bl) : l < ω}.(⇒): Let (bl)l<ω, indiscernible over A, witness dividing. Since ϕ(x , b)does not divide over A in T0, there must be a copy of Kn in Ac(bl)l<ω,where c is an “optimal” solution of {ϕ(x , bl) : l < ω} in T0. Byindiscernibility, this Kn will “project” to the required conditions in Ab.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 16 / 27
Dividing in Tn - Examples
CorollarySuppose A ⊂M and b1, . . . ,bn−1 ∈M\A are distinct. Then the formula
ϕ(x , b) :=n−1∧i=1
xRbi
divides over A if and only if b is not n-bound to A.
Corollary
Let A ⊂M and ϕ(x , y) ∈ LRA . Suppose b ∈M\A such that ϕ(x , b) is
consistent and divides over A. Define
Rϕ = {u ∈ Ab : ∃i ϕ(x , b) B xiRu} ∪ {xi : ∃u ∈ Ab, ϕ(x , b) B xiRu}.
Then |Rϕ| ≥ n and |b ∩ Rϕ| > 1.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 17 / 27
Recall: | in T0
TheoremLet A,B, c ⊂M |= T0. Then
c |A
B ⇔ c | dA
B ⇔ c ∩ B ⊆ A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 18 / 27
| in Tn
Lemma
Let A ⊂M |= Tn and p ∈ Sm(A). If p ` x = b for some b ∈M thenb ∈ A. If p ` xRb for some b ∈M, then either b ∈ A or p ` x = a forsome a ∈ A.
TheoremLet A,B, c ⊂M |= Tn. Then
c |A
B ⇔ c | dA
B ⇔ c ∩ B ⊆ A and for all b ∈ B\A, b iseither n-bound to A or not n-bound to Ac.
CorollaryLet A ⊂M |= Tn. If π(x) is a partial type over A, then π(x) does notfork over A.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 19 / 27
Forking in Tn
Lemma
Suppose (bl)l<ω is an indiscernible sequence in M |= Tn such thatl(b0) = 4 and b0 is K2-free. Then either there are i < j such that{bl
i ,blj : l < ω} is K2-free, or
⋃l<ω bl is not K3-free.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 20 / 27
Forking in Tn
TheoremLet M |= Tn and Ab ⊂M such that l(b) = 4, bRA, A ∼= Kn−3, and b isK2-free. For i 6= j , let
ϕi,j(x ,bi ,bj) = xRbi ∧ xRbj ∧∧a∈A
xRa,
and set ϕ(x , b) :=∨i 6=j
ϕi,j(x ,bi ,bj).
Then ϕ(x , b) forks over A but does not divide over A.
A ∼= Kn−3
b1 b2 b3 b4
ϕ(x , b) = “xRA ∧ |{bi : xRbi}| = 2”
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 21 / 27
Higher Arity GraphsNow suppose R is a relation of arity r ≥ 2. An r -graph is anL-structure satisfying the following sentence
∀x1 . . . ∀xr
R(x1, . . . , xr )→
∧i 6=j
xi 6= xj ∧∧σ∈Sr
R(xσ(1), . . . , xσ(r))
.
So R can be thought of as a collection of r -element subsets of ther -graph.
Let T r0 and T r
n , for n > r , be the natural analogs of T0 and Tn,respectively, to r -graphs.
Call T r0 the theory of the random r -graph, and T r
n the theory ofthe random K r
n -free r -graph, where K rn is the complete r -graph of
size n.T r
n can be thought of as the theory of the unique (countable)Fraısse limit of the class of finite K r
n -free r -graphs.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 22 / 27
Recall - Dividing in T0
Lemma
Let A ⊂M |= T0, b, c ∈M such that c | ∩A
b. If (bl)l<ω isA-indiscernible, with b0 = b, then there is d ∈M with d | ∩
A
⋃l<ω bl
such that d bl ≡A cb for all l < ω.
LemmaLet A ⊂M |= T0 and p(x , y) ∈ S(A) such that xi 6= yj ∈ p(x , y) for alli , j . Suppose p(x , b) is consistent and (bl)l<ω is indiscernible over Awith b0 = b. Then there is a solution c of
⋃l<ω p(x , bl).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 23 / 27
Dividing in T r0
LemmaLet A ⊂M |= T r
0 and p(x , y) ∈ S(A) such that xi 6= yj ∈ p(x , y) for alli , j . Suppose p(x , b) is consistent and (bl)l<ω is indiscernible over Awith b0 = b. Then there is a solution c of
⋃l<ω p(x , bl).
In fact, we can take c to be an optimal solution, in particular ifA0 ⊆ A(bl)l<ω, then for any i1, . . . , ik
M |= R(ci1 , . . . , cik ,A0) ⇔ R(xi1 , . . . , xik ,A0) ∈⋃l<ω
p(x , bl).
Corollary (T r0 is simple.)
Suppose B ⊂M |= T r0 and p ∈ S(B). Then p does not fork over
A = {b ∈ B : ∃i xi = b ∈ p(x)}.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 24 / 27
Dividing in T rn , r > 2
LemmaAssume r > 2. Let A ⊂M |= T r
n and p(x , y) ∈ S(A) such thatxi 6= yj ∈ p(x , y) for all i , j . Suppose p(x , b) is consistent and (bl)l<ω isindiscernible over A with b0 = b. Then there is a solution c of⋃
l<ω p(x , bl).
Corollary (T rn is simple if r > 2.)
Suppose r > 2 and B ⊂M |= T rn and p ∈ S(B). Then p does not fork
over A = {b ∈ B : ∃i xi = b ∈ p(x)}.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 25 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
Dividing in T rn , r > 2
Proof of Lemma.
A
b0
b1
b2
b3
b4
...
c ∈M′ |= T r0 is an optimal
solution to⋃
l<ω p(x , bl).
c
Show Ac(bl)l<ω is K rn -free.
Let K rn∼= W ⊆ Ac(bl)l<ω.
W ∼= K rn
There is ci ∈W ∩ c.
ci
If |l : W ∩ bl 6= ∅}| > 1
u
v
pick W0 ⊆W\{ci ,u, v},with |W0| = r − 3.
W0
R(ci ,u, v ,W0) ∈⋃
l<ω p(x , bl).
|{l : W ∩ bl 6= ∅}| ≤ 1
W ∼= K rn
bm
“W ∼= K rn ” ∈ p(c, bm)
This contradicts M |= ∃xp(x , bm).
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 26 / 27
References
B. Kim & A. Pillay, Simple theories, Annals of Pure and AppliedLogic 88 (1997) 149-164.
D. Marker, Model Theory: An Introduction, Springer, 2002.
K. Tent & M. Ziegler, A Course in Model Theory, Cambridge, 2012.
Gabriel Conant (UIC) Forking and Dividing in Random Graphs April 29, 2012 27 / 27