Dependence and Isolated Extensions

In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an"elementary \phi-extension"(see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.


Introduction
There is a characterization of the stability of a formula ϕ(x; y) in terms of the definability of all ϕ-types. A partitioned formula ϕ(x; y) is stable if and only if all ϕ-types are definable by a formula over their domain [Sh]. We create an analogous result for dependent formulas (that is, formulas without the independence property, sometimes referred to as "NIP" formulas). Since dependence is a strictly weaker notion than stability, we cannot hope to have definability of ϕ-types over their domain for general dependent formulas, ϕ. However, we change the conclusion slightly, in two separate ways, and get a characterization of dependent formulas.
First, we weaken the requirement that a ϕ-type p be definable over dom (p). Instead, we take a model M containing dom(p), take an elementary extension (N; B) of the pair structure (M; dom(p)), and demand that p be definable over B. Second, we strengthen the method by which the ϕ-type p is definable. Instead of being merely definable over this expanded set B, we demand that there exists an extension of p to a ϕ-type p ′ such that dom(p ′ ) ⊆ B and p ′ is ϕ-isolated. From all of this, we construct an analogous result to the characterization of stable formulas, the Isolated Extension Theorem (Theorem 2.4 below). The proof of this theorem is loosely based on a paper by Shelah [Sh900].
In Section 2 we discuss definitions, state the main theorem, and list some consequences of that theorem. The main theorem, Theorem 2.4, is proved in Section 3. Finally, in the Section 4, we discuss the implications of this theorem to the stable case. Even in the stable case, Theorem 2.4 provides new information. Date: November 6, 2009. Special thanks to Chris Laskowski.

Definitions and The Isolated Extension Theorem
Fix a complete, first-order theory T in a language L. We include the case where L is multisorted, so we need to keep track of the sorts of variables. For convenience, if ψ(x) is any formula, then let ψ(x) 0 = ¬ψ(x) and let ψ(x) 1 = ψ(x).
For the first three definitions, fix ϕ(x; y) a partitioned formula of L. By a ϕ-type, we mean a consistent set of formulas p(x) = {ϕ(x; b) s(b) : b ∈ B} for some set B of elements of the same sort as y and some s ∈ B 2 (the set of functions from B to 2 = {0, 1}). We say that dom(p) = B and the space of all ϕ-types over B is denoted For any model M |= T , for any a from M and any B a set of elements of the same sort as y from M, let tp ϕ (a/B) be the ϕ-type over B given by: The above notions can be defined for sets of formulas Γ(x; y) (instead of a single formula) in the obvious way. Throughout this section, when we mention a ϕ-type over B, look at tp ϕ (a/B), or consider the set S ϕ (B), we want B to be a set of elements of the same sort as y (that is, if y = (y 0 , ..., y n−1 ), then B is a set of n-tuples b = (b 0 , ..., b n−1 ) such that b i is of the same sort as y i for all i < n). In Section 3 when we consider ∆-types, we will alter this notation slightly for simplification. When we consider the set of formulas ∆(y; z 0 , ..., z n−1 ) where all the z i 's are of the same sort and B is a set of elements of that sort, we will abuse notation and say that a ∆-type is over B when it is actually over B n and we will write tp ∆ (c/B) when we mean tp ∆ (c/B n ).
Definition 2.1. We say that a set B of elements of the same sort as y is ϕ-independent if, for all s ∈ B 2, the set of formulas {ϕ(x; b) s(b) : b ∈ B} is consistent. We say that ϕ has independence dimension n < ω, denoted ID(ϕ) = n, if n is maximal such that, for some (equivalently any) model M |= T , there exists a set B of elements of the same sort as y from M with |B| = n such that B is ϕ-independent. If such an n exists, then we say that ϕ is dependent. If no such n exists, then we say that ϕ is independent.
Notice that when B is finite, B is ϕ-independent if and only if |S ϕ (B)| = 2 |B| .
Definition 2.2. We say that a ϕ-type p(x) is ϕ-isolated if there exists a finite ϕ-subtype p 0 (x) ⊆ p(x) such that p 0 (x) ⊢ p(x). We say that a formula ψ(x) is a ϕ-formula if it is of the form ψ(x) = i<n ϕ(x; b i ) s(i) for some n < ω, some elements b i of the same sort as y, and some s ∈ n 2.
We see that a ϕ-type p(x) is ϕ-isolated if and only if there exists a ϕ-formula, ψ(x) over dom(p) such that p(x) is equivalent to ψ(x). This ϕ-formula is simply the conjunction of the finite ϕ-subtype p 0 (x) given in Definition 2.2. Now we are ready to state the main theorem of the paper. We will give the proof in Section 3 below.
Theorem 2.4 (The isolated extension theorem). For any partitioned formula ϕ(x; y), the following are equivalent:

Moreover, if the above conditions hold, we can choose
We remark on some consequences of the theorem.
Definition 2.5. Fix a partitioned formula ϕ(x; y), a ϕ-type p(x), and a formula ψ(y). We say that ψ defines p if, for all b ∈ dom(p), ϕ(x; b) ∈ p(x) if and only if ψ(b) holds. We say that ψ ϕ-defines p if it defines p and it is of the form ψ(y) = ∀x(γ(x) → ϕ(x; y)) for some ϕ-formula γ(x).
Merely requiring that a ϕ-type has a defining formula has no content. Indeed, for any type p ∈ S ϕ (B), p is defined by the formula ϕ(a; y) for any realization a of p. The strength of having a defining formula is to have one with a controlled domain, preferably over dom(p). It is known, for example, that for stable formulas ϕ, all ϕ-types p have a defining formula over dom(p) [Sh], but, when dom(p) is an arbitrary set, it does not necessarily have a ϕ-defining formula over dom(p).
Notice that if p is ϕ-isolated, then p has a ϕ-defining formula ψ over dom(p). Namely, take the ϕ-formula γ over dom(p) such that p(x) is equivalent to γ(x) and let ψ(y) = ∀x(γ(x) → ϕ(x; y)). It is clear that if ψ ϕ-defines p, then ψ defines p, but the converse does not necessarily hold. We immediately get the following corollary to Theorem 2.4.
Notice that Corollary 2.6 is, on the one hand, stronger than standard definability of types for stable formulas, and, on the other hand, weaker. We get that, for dependent formulas ϕ, ϕ-types are not only definable, but ϕ-definable. However, the formula doing the defining is not over dom(p), but over B ′ for some (N; B ′ ) (M; dom(p)).
As in the stable case, this ϕ-definability of types leads to a notion of stable embeddability.

The Proof of the Isolated Extension Theorem
To aid notation, assume that the length of x and the length of y is 1. Other than having more complicated notation, the general case is identical.
First, to show (ii) implies (i), we will exhibit the contrapositive. Assume then that ϕ(x; y) is independent. By compactness, there exists a model M with an infinite ϕ-independent set B. Let (N; B ′ ) (M; B). By elementarity, it follows that all finite subsets of B ′ are ϕ-independent. Let p ′ be any extension of p to a ϕ-type such that dom(p ′ ) ⊆ B ′ . Fix any finite subtype p 0 (x) ⊆ p ′ (x). Now, for any finite ϕ-type p This shows that no elementary ϕ-extension of p is ϕ-isolated. Therefore, (ii) implies (i).
To show (i) implies (ii), we will first show that the following proposition holds: Proposition 3.1. For any dependent formula ϕ(x; y) in a theory T , for any model M |= T , for any partial type Θ(y) over ∅, and for any B ⊆ Θ(M), there exists N M and C ⊆ Θ(N) with |C| ≤ 2 · ID(ϕ) and an extension p ′ (x) ∈ S ϕ (B ∪ C) of p(x) that is ϕ-isolated.
Fix ϕ(x; y) a dependent formula in a theory T and Θ(y) any partial type over ∅. Let n = ID(ϕ), the independence dimension of ϕ(x; y). Fix M |= T , N M sufficiently saturated, B ⊆ Θ(M), and p(x) ∈ S ϕ (B). If B is finite, p is already isolated, so assume that B is infinite. Define a set of formulas ∆(y; z 0 , ..., z n−1 ) as follows: (3) ∆(y; z 0 , ..., z n−1 ) = ∃x ϕ(x; y) t ∧ i<n ϕ(x; z i ) s(i) : t < 2, s ∈ n 2 We will now define the notion of a good configuration. This will end up allowing us to build up the external C in at most ID(ϕ) steps (adding two elements at a time).
Definition 3.2. A good configuration of p of size K is a sequence C = {c i,t : i < K, t < 2} such that the following conditions hold: (i) c i,t |= Θ(y) for all i < K, t < 2; (ii) p(x) ∪ {ϕ(x; c j,t ) t : j < K, t < 2} is consistent; and (iii) For all s ∈ K 2, all j < K, c j,0 and c j,1 have the same ∆-type over B ∪ {c i,s(i) : i = j}.
If C is a good configuration of p of size K, then let p C (x) = p(x) ∪{ϕ(x; c j,t ) t : j < K, t < 2}.
The first thing to note is that these good configurations are used to extend the type p in a very specific way. These could, a priori, be arbitrarily large. However, the fact that ϕ is dependent forces good configurations to be of bounded size.
Lemma 3.3. If C = {c i,t : i < K, t < 2}, is a good configuration of p of size K, then K ≤ n = ID(ϕ).
Proof. Suppose not, i.e. K > n. Now, for each s ∈ n+1 2, notice that (4) |= ∃x i<n+1 ϕ(x; c i,s(i) ) s(i) because {ϕ(x; c i,s(i) ) s(i) : i < n + 1} is a consistent type. Now, notice that, for any j ≤ n, because c j,0 and c j,1 have the same ∆-type over {c i,0 : i < j} ∪ {c i,s(i) : j < i < n + 1}. Starting with (4), then using (5) and induction, we get that: But this holds for any s ∈ n+1 2. This contradicts the fact that n = ID(ϕ). Now that we have good configurations, we need a sufficient condition for taking a good configuration and building a larger one out of it. Clearly any new d 0 and d 1 we would like to add on must realize Θ and must be so that ¬ϕ(x; d 0 ) ∧ ϕ(x; d 1 ) is consistent with p C (x). However, the third condition for a good configuration is a bit tricky. Not only do d 0 and d 1 have to have the same ∆-type over B ∪ {c i,s(i) : i < K}, but also each c j,0 and c j,1 have to have the same ∆-type over B ∪ {c i,s(i) : i = j} ∪ {d t }. We now give a sufficient condition for being able to add on to good configurations.
Lemma 3.4. If C = {c i,t : i < K, t < 2} is a good configuration of p, and there exists d 0 , d 1 such that: Then, C ∪ {d 0 , d 1 } is a good configuration of p (of size K + 1).
Proof. Clearly all conditions for C ∪ {d 0 , d 1 } to be a good configuration of p are met except perhaps the condition that c j,0 and c j,1 have the same ∆-type over B ∪ {c i,s(i) : i = j} ∪ {d t } for all s ∈ K 2, t < 2. So suppose this fails, and fix the s ∈ K 2 and t < 2 where this fails.
Then there exists δ either an element of ∆ or the negation of an element of ∆ such that N |= δ(c j,0 , e) ∧ ¬δ(c j,1 , e) for some e from B ∪ {c i,s(i) : i = j} ∪ {d t }. Since c j,0 and c j,1 have the same ∆-type over B ∪ {c i,s(i) : i = j}, we must have that e = d t ⌢ e ′ for some e ′ from B ∪ {c i,s(i) : i = j}. Therefore, we get that: By condition (iv) of the hypothesis, there exists b ∈ B such that: But, as b ⌢ e ′ is from B ∪ {c i,s(i) : i = j}, this contradicts the fact that c j,0 and c j,1 have the same ∆-type over B ∪ {c i,s(i) : i = j}.
Fix C a maximal good configuration of p, so p C (x) is a ϕ-type over B ∪ C. Let s(x) be any extension of p C (x) to a complete type over B ∪ C. Define r s (y) as follows: Lemma 3.5. r s is not finitely satisfied in B.
We will now show how the non-finite-satisfiability of r s in B leads to a formula definition of p C (x).
Lemma 3.6. For any C a maximal good configuration of p and any s( Proof. Consider r s as given above. Then, since r s is not finitely satisfiable in B, there exists m < ω and ψ ℓ (x) ∈ s(x) for each ℓ < m such that, for all b ∈ B, N |= ¬∃x(ϕ(x; b) t ∧ ψ ℓ (x)) for some ℓ < m and some t < 2 (notice here that b |= Θ(y) for all b ∈ B, so that the formulas in Θ(y) ⊆ r s (y) are always realized in B). Let γ(x) be defined as follows: Since s is closed under conjunction, s extends p C , and ψ ℓ (x) ∈ s(x), we get that γ(x) ∈ s(x).
Now that we have a formula definition for p C (x) for each s ∈ S(B ∪ C), we will see that a single formula is equivalent to p C (x) using compactness. After that, we will show that this means a finite ϕ-subtype of p C (x) is equivalent to the whole of p C (x).
Lemma 3.7. If C = {c i,t : i < K, t < 2} is a maximal good configuration of p, then there exists a formula ψ(x) over B ∪ C such that ψ(x) is equivalent to p C (x).
Proof. For each such s(x) ∈ S(B ∪ C) extending p C (x), define γ s (x) to be the formula such that γ s (x) ∈ s(x) and γ s (x) ⊢ p C (x) as given in Lemma 3.6.
Lemma 3.8. If C = {c i,t : i < K, t < 2} is a maximal good configuration of p, then there exists a finite ϕ-subtype p 0 (x) ⊆ p C (x) so that p 0 (x) ⊢ p C (x).
Proof. First let ψ(x) be a formula over B ∪ C that is equivalent to p C (x), given by Lemma 3.7. Then consider {¬ψ(x)} ∪ p C (x), a partial type over B ∪ C. This is clearly inconsistent.
On the issue of uniformity, the results of Theorem 2.4 differ strongly from the standard definability of ϕ-types in the stable case. In the case where ϕ is stable, we can use a compactness argument to get a uniform definition of ϕ-types. Note, however, that this uniform definition is not necessarily a ϕ-definition. One cannot, in general, get a uniform ϕ-definition of all ϕ-types, even in the case where ϕ is stable.
As an example, let T be the theory, in the language L = {E} with a single binary relation E, stating that E is an equivalence relation with infinitely many E-equivalence classes all of infinite size. This theory is certainly stable, and even ℵ 0 -stable. Fix M |= T and let B ⊂ M be a set containing one element from one class, two from another, three from a third class, and so on. Finally, let ϕ(x; y, z, w) be the formula given by: (15) ϕ(x; y, z, w) = [(z = w → x = y) ∧ (z = w → E(x, y))] (so ϕ encodes the two formulas "x = y" and E(x, y) into a single formula). Now let n ∈ ω be arbitrary and let a ∈ M − B be in the E-equivalence class with exactly n elements of B in it; call this class [a] E . Finally, let p n (x) = tp ϕ (a/B). Now, for any (N; B ′ ) (M; B), notice that the E-equivalence class with exactly n elements from B still has exactly n elements from B ′ , so [a] E ∩ B ′ = [a] E ∩ B. However, this shows that any ϕ-extension of p n to some p ′ with dom(p ′ ) ⊆ B ′ is ϕ-isolated only by a finite subtype whose domain contains [a] E ∩ B (this is because we need the full set [a] E ∩ B to say that x = b for each b ∈ ([a] E ∩ B) yet E(x, b) for some (all) b ∈ ([a] E ∩ B)). As |[a] E ∩ B| = n and n < ω was arbitrary, we see from this example that there is no uniform bound on the size of the ϕ-isolating ϕ-subtype of the elementary ϕ-extension given by Theorem 2.4, even in the stable case.