1 1 1. Stability Stability is fundamentally a local hypothesis, indeed a combinatorial hy- pothesis on a bipartite graph. A bi-partite graph is just a triple of sets G = (G1;G2; I) with I G1 G2; one thinks of I as relating elements of ⊂ × G1 to elements of G2. An example is Ln, where G1 and G2 are two copies of 1; : : : ; n , and I is the graph of the order relation i < j. f g Definition 1.1. A bi-partite graph is stable if for some n, G does not embed the graph Ln Definition 1.2. A Γ-formula a formula equivalent to a quantifier-free formula, possibly with additional parameters. A set C is Γ-algebraically-closed if for every Γ-formula E over C defining an equivalence relation with finitely many classes, each class of E is coded by an element of C.A Γ-type over C is a maximal consistent collection of Γ-formulas over C.A Γ-type p over Γ is clearly determined by its restriction to atomic formulas; we will say p is definable if there exists a formula α(y) such that xIy p iff α(y) (or dually; we will write 2 xIy or yIx interchangeably.) If p is definable, the defining formula is unique up to logical equivalence, and will be denoted (dpx)(xIy). Atomic types over Γ are assumed consistent with Th(Γ) (equivalently, with the universal theory of Γ.) Lemma 1.3. The following are equivalent: i) Γ is stable ii) For every countable Γ0 satisfying the universal theory T of Γ, there are 8 countably many atomic types over Γ0 consistent with the universal theory of Γ0. 1Adapted from: Model theory and applications, 151{212, Quad. Mat., 11, Aracne, Rome, 2002. 2 iii) There is no sequence (aibi : i < !) in an elementary extension of Γ such that aiIbj iff i < j: Proof { The equivalence of (i),(iii) is immediate, and (i) (ii) is a : !: standard excerices in compactness. (i) (ii) follows from 1.6(a). ! Definition 1.4. Let N be a structure for a language L0, and let a : N U be ! a function. Let Θ be a collection of formulas. For θ = θ(x1; : : : ; xn) Θ, let n 2 a∗(θ) = (c1; : : : ; cn) N : U = θ(a(c1); : : : ; a(cn)). f 2 j We say a is indiscernible for (N; Θ) if for any θ Θ, a∗(θ) is definable in 2 N. Main case: the structure on N consists just of a linear ordering, or no structure at all; and Θ is the collection of all formulas. In these cases we say that a is an indiscernible sequence, respectively an indiscernible set. Remark 1.5 - In Lemma 1.3 (iii), we can take the sequence to be indiscernible. Proof { It is clear that any infinite subsequence will have the same prop- erty. If Θ is a finite set of formulas, it follows from Ramsey's theorem that any sequence (ai : i N) has an infinite (<; Θ)-indiscernible subsequence. By com- 2 pactness we can obtain an indiscernible sequence with the same property. Theorem 1.6. Let Γ be a stable bi-partite graph a) (Definability) Every atomic type over Γ is definable, by a Γ- formula. b) (Existence of nonforking extension.) Let C Γ, p a Γ-type over C. Then ⊆ there exists a Γ-type p∗ over Γ such that p∗ p , and p∗ is definable over ⊇ acl(C). c) (Symmetry). Let C be Γ-algebraically closed. Suppose pi(xi) are Γ-types over Γ, pi(xi) implies xi Γi, and p1, p2 are definable over acl(C). Let 2 ai realize pi C. Then a1Ix2 p2 iff x1Ia2 p1. j 2 2 d) (Uniqueness.) Let C be Γ-algebraically closed. Then the nonforking ex- tension p∗ in (b) is unique. Hence it is defined over C. 3 e) (Forking characterization) Let p(x) be a Γ-type over C = aclΓ(C), p∗ its non-forking extension, and suppose xIb = p∗. Then there exist conjugates 2 bi of b (i !) such that p(x) xIbi : i ! is inconsistent. 2 [ f 2 g f) (Finiteness and conjugacy). The set X(q) of nonforking extensions of a Γ-type q over is finite. There exists a 0-definable, Γ-definable equiva- ; lence relation Eq splitting q into finitely many classes (all realizing the same type), such that in each Eq class there is a unique element of X(q). If G is any group of automorphisms of Γ fixing C and transitive on the Eq-classes, then G is transitive on X(q). Proof { For the proofs Γ may be assumed countable, and further may be replaced by a somewhat saturated elementary extension: enough so that an (imaginary) element is definable over a finite set C iff it has no proper conju- gates over C. We will avoid using local rank, except in quoting the following result from elementary topology: if X is a countable compact Hausdorff space then there exists a finite closed subspace X0 invariant under every automor- phism of X. (Namely the points left over at the end of a Cantor-Bendixon analysis.) (a) Note that there can be no (ai; bi; ci) with tp (an= ci : i < n ) = tp (bn= ci : at f g at f i < n ), but anIci (bnIci) for n < i. For by Ramsey's theorem we may g ≡ : assume either bn ci whenever i < n, or (anIci) whenvever i < n; and an j : infinite sequence of either (aici) or (bici) will contradict stability. Given an atomic type p over Γ, pick ai; bi; ci Γ such that cn realizes p ai; bi : i < n . 2 jf g At some point it becomes impossible to choose an; bn realizing the same atomic type over ci : i < n but such that (anIx p) (bnIx p). Clearly p is f g 2 ≡ : 2 atomically definable over ci : i < n . f g b) The space X of Γ-types over Φ is compact, Hausdorff and by 1.3(ii) also countable. It follows that X has an Aut(X)-invariant finite subset X0. Let p X0. Then p has finitely many conjugates over C, hence the p-definition of 2 I has finitely many such conjugates; so it is over acl(C). c) Let qi = pi C. Let Xi be the set of nonforking extensions of qi, in the sense j of (b). The equivalence relation Ei defined by: xEix0 iff for each p Xi, 2 (dpy)(xIy) = (dpy)(xIy) is defined over C, has finitely many classes, and is a Γ-formula by 2(a). Since C is Γ-algebraically closed and q3 i is Γ-complete, − 4 every element of q3 i lies in a single class of Ei. Hence if a, a0 realize q1 and b − realizes p2 a; a0 then aIb iff a0Ib; and dually. Thus if the required statement jf g fails, then we have: (∗) whenever ai realizes qi,(a1Ix2) p2 iff (x1Ia2 p1). 2 : 2 Pick ai; bi Γ such that an realizes p1 a1; : : : ; an 1; b1; : : : ; bn 1 and bn re- 2 jf − − g alizes p2 a1; : : : ; an; b1; : : : ; bn 1 . Then by (∗) aiIbn iff (anIbi) for i < n. jf − g : This contradicts the assumption of stability. d) Suppose p1, p10 are both definable over acl(C) and extend p. Pick any ele- ment a, and let q be an acl(C)-definable extension of tp(a=C). Let b realize p. Then by (c), xIa p1 iff bIy q iff xIa p0 . So p1 = p0 . 2 2 2 1 1 e) Let q be a nonforking extension of tp(b=C). Let bn realize q b0; : : : ; bn 1 . jf − g Suppose p(x) xIbi : i ! is consistent; let r0 be a complete type over [ f 2 g acl(C b0; b1;::: ) extending it, and let r be a nonforking extension of r0. r [ f g is definable over B = acl(C b0; : : : ; bn 1 ) for some n. Let a realize r B. By [ f − g j (c) we have: i) xIbn r iff aIy q. 2 2 ii) xIb p∗ iff aIy q. 2 2 But xIb = p∗ by assumption, while so xIbn r0 r a contradiction. 2 2 ⊆ f) By (C) there is a 1-1 correspondence between X(q) and the set Y (q) of ex- tensions of q to Γ-types over aclΓ( ). So we may consider Y (q) instead of X(q). ; Let q0 Y (q), and let p be any complete type extending q. Then p q0 is con- 2 [ sistent: otherwise let φ q0, φ p; let φ0 be the disjunction of all conjugates 2 : 2 of φ; then φ0 p but φ0 q0 and φ0 is over , so φ0 q, a contradiction. : 2 2 ; 2 Now given q1; q2 Y (q), let ai realize qi p; then for some automorphism 2 [ σ we have σa1 = a2, and σq1 = tp (σa1=aclΓ( )) = q2. This shows (using Γ ; our saturation assumption) that Aut(G) is transitive on Y (q) and hence on X(q). Since by (a) an element of X(q) is coded by a single imaginary element, we have a conjugacy class of imaginary elements, each of which is algebraic over ; it follows the class is finite. Thus Y (q) is finite. Define: a1Eqa2 iff ; tp (a1=aclΓ( )) = tp (a2=aclΓ( )); then by the finiteness Eq is definable. The Γ ; Γ ; last two statements are immediate. Lemma 1.7. Suppose (Γ1; Γ2; I) and (Γ1; Γ0 ; I0) are stable. Let Γ∗ = Γ2 Γ0 , 2 2 × 2 and let I∗ be any Boolean combination of I and I0.