Countable imaginary simple unidimensional theories are supersimple

Ziv Shami Tel Aviv University

November 10, 2008

Abstract

We prove that a countable simple unidimensional theory that eliminates hyperimaginaries is supersimple. This solves a problem of Shelah in the more general context of simple theories under weak assumptions.

1 Introduction

The notion of a unidimensional theory already appeared (in a different form) in Baldwin-Lachlan characterization of ℵ_{1}-categorical theories; a countable theory is ℵ_{1}-categorical iff it is ω-stable and unidimensional (equivalently, T has no Vaught pairs and is ω-stable). Later, Shelah defined a unidimensional theory to be a stable theory T in which every two |T |^{+}-staurated models of the same power are isomorphic, and proved that in the stable context a theory is unidimensional iff every two non-algebraic types are non-orthogonal. A problem posed by Shelah was whether any unidimensional stable theory is superstable. This was answered positively by Hrushovski around 1986 first in the countable case [H0] and then in full generality [H1]. Taking the right hand side of Shelah characterization of unidimensional stable theories seems natural for the simple case. Shelah’s problem extended to this context seems much harder. In [S3] it was observed that a small simple unidimensional theory is supersimple. Later, Pillay [P] gave a positive answer for countable imaginary simple theories with wnfcp (the weak non finite cover property) building on the arguments in [H0] and using some machinary from [BPV]. Then using the result on elimination of ∃^{∞} in simple unidimensional theories [S1] completed his proof for countable imaginary low theories. In this paper we prove the result for any countable imaginary simple theory. In general, the main idea is the dividing line ”T is essentially 1-based” which means that every type is coordinatized by essentially 1-based types in the sense of the forking topology (the τ^{f}-topology). In case T is not essentially 1-based we prove there is an unbounded τ^{f}-open set of finite SU-rank (possibly with no finite bound); this is a general dichotomy for countable imaginary simple theories. Then one applies the result in [S2] and a property (PCFT) proved in the current paper to conclude T is supersimple. If T is essentially 1-based the proof first reduces the problem to finding a type-definable τ^{f}-open set of bounded finite SU_{s}-rank (the foundation rank with respect to forking with stable formulas). In order to show the existence of such sets, we introduce the notion of a τ̃_{st}^{f}-set and prove a theorem saying that in any simple theory in which the extension property is first-order any unbounded τ̃_{st}^{f}-set contains an unbounded type-definable τ^{f}-open set (over parameters). Then, in the proof of the main result we show how unidimensionality of T implies there is a τ̃_{st}^{f}- set that by a more precise version of the theorem is in fact a type-definable τ^{f}-open set over some finite set and is of bounded finite SU_{s}-rank.

2 Preliminaries

We will assume basic knowledge of simple theories as in [K1],[KP],[HKP]. A good text book on simple theories that covers much more is [W]. We recall here some other definitions and facts relevant for this paper. In this section T will be a simple theory and we work in C^{eq}, the monster model of T^{eq}.

2.1 Interaction

For the rest of this section let P be an A-invariant set of small types and p ∈ S(A). We say that p is (almost-) P-internal if there exists a realization

a of p and there exists B ⊇ A with \begin{aligned}a & ⌣ | & B \\ & A \end{aligned} such that for some tuple

c of realizations of types in P over B we have a ∈ dcl(B, c) (respectively, a ∈ acl(B, c)). We say that p is analyzable in P if there exists a sequence I = ⟨a_{i}|i ≤ α⟩ ⊆ dcl^{eq}(a_{α}A) where a_{α} ⊧ p and tp(a_{i}/A ∪ {a_{j}|j < i}) is P-internal for every i ≤ α. We say that p is foreign P if for every B ⊇ A and

a ⊧ p with \begin{aligned}a & ⌣ | & B \\ & A \end{aligned} and a realization c of a type in P over B, \begin{aligned}a & ⌣ | & c \\ & B \end{aligned} .

Also, recall that p ∈ S(A) is said to be orthogonal to some q ∈ S(B) if for every C ⊇ A ∪ B, for every p̄ ∈ S(C), a non-forking extension of p , and every q̄ ∈ S(C), a non-forking extension of q, for every realization a of p̄ and

realization b of q̄, \begin{aligned}a & ⌣ | & b \\ & C \end{aligned} .

We say that T is imaginary (or has elimination of hyperimaginaries) if for every type-definable over ∅ equivalence relation E on a complete type p (of a possibly infinite tuple of elements), E is equivalent on p to the intersection of some definable equivalence relations E_{i} ∈ L.

Fact 2.1 Let T be an imaginary simple theory. Then 1)If p is not foreign to P, then for a ⊧ p there exists a′ ∈ dcl(Aa)\acl(A) such that tp(a′/A) is P-internal.

2) p is analyzable in P iff every non-algebraic extension of p is non-foreign to P.

An easy fact we will be using is the following.

Fact 2.2 Assume tp(a_{i}) are P-internal for i < α. Then tp(⟨a_{i}|i < α⟩) is P-internal.

An important characterization of almost-internality is the following fact [S0, Theorem 5.6.] (a similar result obtained independently in [W, Proposition 3.4.9]).

Fact 2.3 Let p ∈ S(A) be an amalgamation base and let U be an ∅-invariant set. Suppose p is almost-U-internal. Then there is a Morley sequence ā in p and there is a definable relation R(x, ȳ, ā) (over ā only) such that, for every tuple c̄, R(C, c̄, ā) is finite and for every a′ realizing p, there is some tuple c̄ from U such that R(a′, c̄, ā) holds.

T is said to be unidimensional if whenever p and q are complete non-algebraic types, p and q are non-orthogonal. From Fact 2.3 and Fact 2.1 it is easy to deduce the following (using compactness).

Fact 2.4 Let T be a simple theory. Let p ∈ S(∅) and let θ ∈ L. Assume p is analyzable in θ^{C}. Then p is analyzable in θ^{C} in finitely many steps. In particular, if T is an imaginary simple unidimensional theory and there exists a non-algebraic supersimple definable set (i.e. a definable set with global Drank), then T has finite SU-rank i.e., every complete type has finite SU-rank (in fact, for every given sort there is a finite bound on the SU-rank of all types in that sort).

2.2 The forking topology

Definition 2.5 Let A ⊆ C. An invariant set U over A is said to be a basic τ^{f}-open set over A if there is ϕ(x, y) ∈ L(A) such that

U = {a|ϕ(a, y) forks over A}.

Note that the family of basic τ^{f}-open sets over A is closed under finite intersections, thus form a basis for a unique topology on S_{x}(A).

Definition 2.6 We say that the τ^{f}-topologies over A are closed under projections (T is PCFT over A) if for every τ^{f}-open set U(x, y) over A the set ∃yU(x, y) is a τ^{f}-open set over A. We say that the τ^{f}-topologies are closed under projections (T is PCFT) if they are over every set A.

We will make an essential use of the following facts from [S2].

Fact 2.7 Let U be a τ^{f}-open set over a set A and let B ⊇ A be any set. Then U is τ^{f}-open over B.

Fact 2.8 Let U be an unbounded τ^{f}-open set over some set A. Assume U has bounded finite SU-rank. Then there exists a set B ⊇ A and θ(x) ∈ L(B) of SU-rank 1 such that θ^{C} ⊆ U ∪ acl(B).

The following theorem ([S2], Theorem 3.11]) generalizes Fact 2.4 but at the price of PCFT.

Fact 2.9 Assume T is a simple theory with PCFT. Let p ∈ S(A) and let U be a τ^{f}-open set over A. Suppose p is analyzable in U. Then p is analyzable in U in finitely many steps.

3 Unidimensionality and PCFT

We show that if T is any simple theory in which the extension property is firstorder (more generally, if T is semi-PCFT) then T is PCFT. Recall that the extension property is first-order for T iff for every formulas ϕ(x, y), ψ(y, z) ∈ L the relations Q_{ϕ,ψ} defined by: for all a, Q_{ϕ,ψ}(a) iff ”ϕ(x, b) does not divide over a for every b ⊧ ψ(y, a)” are type-definable (see Definition 3.9, part 4). In this section T is assumed to be simple.

Definition 3.1 We say that T is semi-PCFT over A if for every formula ψ(x, yz) ∈ L(A) the set {a| ψ(x, ab) forks over Aa for some b} is τ^{f}-open over A.

Lemma 3.2 If the extension property is first-order then the extension property is first-order over every set A. Thus if the extension property is firstorder, then T is semi-PCFT over every set A.

Proof: First, note that the type-definability of the Q_{ϕ,ψ}-s implies that the κ-extension property is first-order for every κ ≥ |T |^{+}. Now, if the κextension property is first-order for every κ ≥ |T |^{+} then, given a (|T |+|A|)^{+}- saturated model M^{∗} = (M_{A}, P (M_{A})) of the common theory of all pairs of T_{A}(= T h(C, a)_{a∈A}) satisfying the extension property, M^{∗}|L_{P} is also a model of the common theory of all pairs of T satisfying the (|T | + |A|)^{+}-extension property (indeed, every pair of T satisfying the extension property has an elementary extension which is sufficiently saturated and thus has an expansion to a pair of models of T_{A}). Thus, since we assume the κ-extension property is first-order for every κ ≥ |T |^{+}, M^{∗}|L_{P} has the (|T |+|A|)^{+}-extension property, and thus M^{∗} has the extension property.

Lemma 3.3 Assume T is semi-PCFT over A. Then T is PCFT over A.

Proof: We may clearly assume A = ∅. Let ψ(x, yz) ∈ L. We need to show that Γ, defined by Γ(a) iff ∀b(ψ(x, ab) dnf over ∅) is a τ^{f}-closed set. Let Γ^{∗} be defined by: for all a:

Γ^{∗}(a) iff ⋀_{ϕ(x,y)∈L}[F K_{ϕ}(a) → ∀b(ψ(x, ab) ∧ ¬ϕ(x, a) dnf over a)].

To finish it is sufficient to prove:

Subclaim 3.4 Γ^{∗} is τ^{f}-closed and Γ = Γ^{∗}.

Proof: First, by our assumption Γ^{∗} is τ^{f}-closed. To prove the second part, first assume Γ(a). Then for any b there is c such that c ⌣ | ab and ψ(c, ab). Thus Γ^{∗}(a). Assume now Γ^{∗}(a). Let p_{a}^{ind}(x) = ⋀ {¬ϕ(x, a)| ϕ(x, y) ∈ L, F K_{ϕ}(a)}. Let b be arbitrary and let q(x) = p_{a}^{ind}(x) ∧ ψ(x, ab). It is enough to show that q(x) doesn’t fork over a (since any realization of q is independent of a). Indeed, by Γ^{∗}(a), every finite subset of q(x) doesn’t fork over a, so we are done.

Corollary 3.5 Suppose the extension property is first-order in T. Then T is PCFT.

Now we conclude that every unidimensional simple theory is PCFT. Recall the following two facts and their corollary. First, let P^{SU≤1} denote the class of complete types over sets of size ≤ |T |, of SU-rank ≤ 1.

Fact 3.6 (Pillay) Let T be a simple theory that eliminates ∃^{∞}. Moreover, assume every non-algebraic type is non-foreign to P^{SU≤1}. Then the extension property is first-order in T.

Fact 3.7 [S1] Let T be any unidimensional simple theory. Then T eliminates ∃^{∞}.

Corollary 3.8 In any unidimensional simple theory the extension property is first-order.

We give now an easy generalization of Fact 3.6. Let us first fix some notations and definitions. By a pair (M, P^{M}) we mean an elementary pair of models of T, where P is a new predicate symbol. For the rest of this section, by a small type we mean a complete hyperimaginary type in ≤ |T | variables over a hyperimaginary of length ≤ |T |.

Definition 3.9 Let P_{0}, P_{1} be ∅-invariant families of small types.

1) We say that a pair (M, P^{M}) satisfies the extension property for P_{0} if for every L-type p ∈ S(A), A ∈ dcl(M) with p ∈ P_{0} there is a ∈ p^{M} such that

\begin{aligned}a & ⌣ | & P^{M} \\ & A \end{aligned} .

2) Let

T_{Ext,P_{0}} = ⋂{T h_{L_{P}} (M, P^{M})| the pair (M, P^{M}) satisfies the extension property w.r.t. P_{0} }.

3) We say that P_{0} dominates P_{1} w.r.t. the extension property if (M, P^{M}) satisfies the extension property for P_{1} for every |T |^{+}-saturated pair (M, P^{M}) ⊧ T_{Ext,P_{0}}. In this case we write P_{0} ⊵_{Ext} P_{1}.

4) We say that the extension property is first-order for P_{0} if P_{0} ⊵_{Ext} P_{0}. The extension property is first-order if the extension property is first-order for the family of all small types.

For an ∅-invariant family P_{0} of small types we say that P_{0} is extensionclosed if for all p ∈ P_{0} if p̄ is any extension of p to a small type, then p̄ ∈ P_{0}.

Lemma 3.10 Let P_{0} be an ∅-invariant family of small types. Assume P_{0} is extension-closed and that the extension property is first-order for P_{0}. Let P^{∗} be the maximal class of small types such that P_{0} ⊵_{Ext} P^{∗}. Then P^{∗} ⊇ An(P_{0}) (where An(P_{0}) is the class of all small types analyzable in P_{0}).

Proof: Note that if the pair (M, P^{M}) satisfies the extension property for the family of ∅-conjugates of tp(b/A) and for the family of ∅-conjugates of tp(a/bA) then (M, P^{M}) satisfies the extension property for the family of ∅- conjugates of tp(ab/A). Thus, since P_{0} is extension-closed (and the extension property is first-order for P_{0}) we conclude that if B is any small hyperimaginary and a_{0}, a_{1}, ..., a_{n} are realizations of some types from P_{0} over B, then if (M, P^{M}) is a |T |^{+}-saturated pair and (M, P^{M}) ⊧ T_{Ext,P_{0}} then (M, P^{M}) satisfies the extension property for the family of ∅-conjugates of tp(a_{0}a_{1}...a_{n}/B). Therefore, P^{∗} ⊇ Int(P_{0}), where Int(P_{0}) denotes the family of small types internal in P_{0}. By induction we conclude that P^{∗} ⊇ An(P_{0}).

Remark 3.11 Note that if T eliminates ∃^{∞} then the extension property is first-order for P^{SU≤1} (this was proved in [V, Proposition 2.15]). Thus Lemma 3.10 implies Fact 3.6.

We conclude:

Theorem 3.12 Let T be any unidimenisonal simple theory. Then T is PCFT.

Corollary 3.13 Let T be an imaginary simple unidimensional theory. Let p ∈ S(A) and let U be an unbounded τ^{f}-open set over A. Then p is analyzable in U in finitely many steps. In particular, for such T the existence of an unbounded supersimple τ^{f}-open set over some set A implies T is supersimple.

Proof: By Theorem 3.12 every unidimensional theory is P CF T. Thus by Fact 2.9 and the assumption that T is imaginary and unidimensional, if U is an unbounded τ^{f}-open set over A, then tp(a/A) is analyzable in U in finitely many steps for every a ∈ C. Thus, if U is supersimple, SU(a/A) < ω for all a ∈ C. Thus T is supersimple.

4 Definability of being in the canonical base

In this section we show that in suitable setting the relation R defined by R(e, a) iff e ∈ acl(Cb(C/a)) is Stone open over C (for a fixed set C).

Definition 4.1 Let C be any set. We say that a set U is a basic τ_{∗}^{f}-open set over C if there exists ψ(x, y, C) ∈ L(C) such that U = {a| ψ(x, aC) forks over a}.

First, we note the following claim:

Claim 4.2 For every e, C, a, we have e ∈ acl(Cb(C/a)) iff for every Morley sequence (C_{i}|i < ω) of Lstp(C/a) we have e ∈ acl(C_{i}|i < ω).

Proof: Right to left follows form the well known fact that Cb(C/a) ∈ dcl(C_{i}|i < ω) for every Morley sequence (C_{i}|i < ω) of Lstp(C/a). For the other direction, assume the right hand side. Let (C_{i}|i < ω · 2) be a Morley sequence of Lstp(C/a). Let e^{∗} = Cb(C/a). Then e^{∗} ∈ bdd(a) and thus clearly (C_{i}|i < ω · 2) is a Morley sequnce of Lstp(C/e^{∗}). In particular, (C_{i}|i < ω) is independent from (C_{i}|ω ≤ i < ω·2) over e^{∗}. By our assumption, e ∈ acl(C_{i}|i < ω) and e ∈ acl(C_{i}|ω ≤ i < ω · 2). Thus e ∈ acl(e^{∗}).

Lemma 4.3 Let C be any set and let (in a given sort) W = {(e, a)| e ∈ acl(Cb(C/a))}. Then W is a τ_{∗}^{f}-open set over C.

Proof: First note that since T is simple, for any two sorts, if x, x′ has the first sort, and y has the second sort, there exists a type-definable relation E_{L}(x, x′, y) such that for all a, a′, b with the right sorts we have E_{L}(a, a′, b) iff Lstp(a/b) = Lstp(a′/b). By Claim 4.2, (e, a) ∉ W iff there exists an aindiscernible sequence (C_{i}|i < ω) which is independent over a with E_{L}(C_{0}, C, a) such that e ∉ acl(C_{i}|i < ω). Let F be the collection of all finite sets of formulas of the form ψ(Y_{0}, ..., Y_{i}, y) ∈ L where each of {Y_{i}}_{i<ω} have the sort of C and y has the sort of a. For Δ ∈ F let n(Δ) be the maximal natural number n such that Y_{n} appears in one of the ψ-s in Δ. Also, for ψ(Y_{0}, ..., Y_{i}, y) ∈ L as above let ψ_{C} = ψ(Y_{0}, ..., Y_{i−1}, C, y) and let i(ψ) = i. For a formula ϕ(x, y, C) ∈ L(C), let F K_{ϕ}^{∗} be the relation over C defined by F K_{ϕ}^{∗}(a) iff ϕ(x, a, C) forks over a. Thus, by compactness (e, a) ∉ W iff

⋀_{Δ∈F}[( ⋀_{ψ∈Δ} F K_{ψ_{C}}^{∗}(a)) → ∃Y_{0}, ...Y_{n(Δ)}Θ(e, a, Y_{0}, Y_{1}, ..., Y_{n(Δ)}, C)].

where Θ is the partial type over C defined by Θ(e, a, Y_{0}, Y_{1}, ..., Y_{n(Δ)}, C) =

E_{L}(Y_{0}, C, a)∧I(Y_{0}, ...Y_{n(Δ)}, a)∧( ⋀_{ψ∈Δ} ¬ψ(Y_{0}, Y_{1}, ..., Y_{i(ψ)}, a))∧e ∉ acl(Y_{0}, Y_{1}, ..., Y_{n(Δ)}).

and where I(Y_{0}, ...Y_{n(Δ)}, a) is the partial type saying Y_{0}, , , , Y_{n(Δ)} is a-indiscernible. Note that the complement of W is an intersection of τ_{∗}^{f}-closed sets over C (note that clearly every Stone-closed set over C is τ_{∗}^{f}-closed over C.)

Proposition 4.4 Let q(x, y) ∈ S(∅) and let χ(x, y, z) ∈ L be such that ⊧ ∀y∀z∃^{<∞}xχ(x, y, z). Then U = {(e, c, b, a)| e ∈ acl(Cb(cb/a))} is relatively Stone-open inside the Stone-closed set

F = {(e, c, b, a)| b ⌣ | a , ⊧ χ(c, b, a), tp(cb) = q}.

Proof: Note that since q ∈ S(∅), it is enough to show that for any fixed c^{∗}b^{∗} ⊧ q the set U^{∗} = {(e, a)| e ∈ acl(Cb(c^{∗}b^{∗}/a))} is relatively stone-open inside

F^{∗} = {(e, a)| b^{∗} ⌣ | a , ⊧ χ(c^{∗}, b^{∗}, a)}.

Now, by Lemma 4.3 we know U^{∗} is a τ_{∗}^{f}-open set over b^{∗}c^{∗}. Thus, for some ψ_{i}(z; x, y, c^{∗}b^{∗}) ∈ L(c^{∗}b^{∗}) (i ∈ I) we have U^{∗} = ⋃_{i} U_{ψ_{i}}^{∗} where U_{ψ_{i}}^{∗} = {(e, a)| ψ_{i}(z; e, a, c^{∗}b^{∗}) forks over ea}.

Subclaim 4.5 For every (e, a) ∈ F^{∗} we have (e, a) ∈ U_{ψ_{i}}^{∗} iff

∀d(ψ_{i}(d; e, a, c^{∗}b^{∗}) → da ⌣̸ | b^{∗} ) ∧ e ∈ acl(a).

Proof: Assuming the left hand side we know e ∈ acl(Cb(c^{∗}b^{∗}/a)), hence

e ∈ acl(a). Let d ⊧ ψ_{i}(z; e, a, c^{∗}b^{∗}). If da ⌣ | b^{∗} , then \begin{aligned}d & ⌣ | & b^{∗} \\ & a \end{aligned} . Since

(e, a) ∈ F^{∗}, c^{∗} ∈ acl(b^{∗}a) implies \begin{aligned}d & ⌣ | & b^{∗}c^{∗} \\ & ea \end{aligned} , contradicting (e, a) ∈ U_{ψ_{i}}^{∗}.

Assume now the right hand side. Let d ⊧ ψ_{i}(z; e, a, c^{∗}b^{∗}). Assume by a way of

contradiction \begin{aligned}d & ⌣ | & b^{∗}c^{∗} \\ & ea \end{aligned} . Since e ∈ acl(a), this equivalent to \begin{aligned}d & ⌣ | & b^{∗}c^{∗} \\ & a \end{aligned} .

Since (e, a) ∈ F^{∗} this is equivalent to da ⌣ | b^{∗} , contradiction.

By Subclaim 4.5 we see that each of U_{ψ_{i}}^{∗} and hence U^{∗} is Stone-open relatively inside F^{∗}.

5 A dichotomy for projection closed topologies

We consider a general family of topologies on the Stone spaces S_{x}(A) that refines the Stone topology and is closed under projections. For any such family of topologies we will prove a dichotomy saying that either there exists an unbounded invariant set U that is open in this topology and is supersimple OR for any SU-rank 1 type p_{0} every type analyzable in p_{0} is analyzable in p_{0} by essentially 1-based types by mean of our family of topologies. In this section T is assumed to be a simple theory with elimination of hyperimaginaries

Definition 5.1 A family

Υ = {Υ_{x,A}| x is a finite sequence of variables and A ⊂ C is small}

is said to be a projection closed family of topologies if each Υ_{x,A} is a topology on S_{x}(A) that refines the Stone-topology on S_{x}(A), this family is invariant under automorphisms of C and change of variables by variables of the same sort, and the family is closed under product by the full Stone space S_{y}(A) (where y is a disjoint tuple of variables) and closed by projections, namely whenever U(x, y) ∈ Υ_{xy,A}, ∃yU(x, y) ∈ Υ_{x,A}.

There are two natural examples of projections-closed families of topologies; the Stone topology and the τ^{f}-topology of a PCFT theory. From now on fix a projection closed family Υ of topologies.

Definition 5.2 1) A type p ∈ S(A) is said to be essentially 1-based over A_{0} ⊆ A, by mean of Υ if for every finite tuple c̄ from p and for every typedefinable Υ-open set U over Ac̄, with the property that a is independent from A over A_{0} for every a ∈ U, the set {a ∈ U| Cb(a/Ac̄) ∉ bdd(aA_{0})} is nowhere dense in the Stone-topology of U. We say p ∈ S(A) is essentially 1-based by mean of Υ if p is essentially 1-based over ∅ by mean of Υ.

2) Let V be an A_{0}-invariant set and let p ∈ S(A_{0}). We say that p is analyzable in V by essentially 1-based types by mean of Υ if there exists a ⊧ p and there exists a sequence (a_{i}| i ≤ α) ⊆ dcl^{eq}(A_{0}a) with a_{α} = a such that tp(a_{i}/A_{0} ∪ {a_{j}|j < i}) is V -internal and essentially 1-based over A_{0} by mean of Υ for all i ≤ α.

Example 5.3 The unique non-algebraic 1-type over ∅ in ACF_{p} is not essentially 1-based by mean of the τ^{f}-topologies for all p.

Remark 5.4 Note that p ∈ S(∅) is essentially 1-based by mean of Υ iff for every finite tuple c̄ from p and for every type-definable Υ-open set U over c̄, there exists a relatively Stone-open non-empty subset χ of U such that

\begin{aligned}a & ⌣ | & c̄ \\ & acl^{eq}(a) ∩ acl^{eq}(c̄) \end{aligned} for all a ∈ χ.

Remark 5.5 Assume d ∈ dcl(c). Then Cb(d/a) ∈ dcl(Cb(c/a)) for all a.

One of the key ideas for proving the main result is the following theorem.

Theorem 5.6 Let T be a countable simple theory that eliminates hyperimaginaries. Let Υ be a projection-closed family of topologies. Let p_{0} be a partial type over ∅ of SU-rank 1. Then, either there exists an unbounded finite-SU-rank Υ-open set over some countable set, or every type p ∈ S(A), with A countable, that is internal in p_{0} is essentially 1-based by mean of Υ. In particular, either there exists an unbounded finite-SU-rank Υ-open set, or whenever A is countable, p ∈ S(A) and every non-algebraic extension of p is non-foreign to p_{0}, p is analyzable in p_{0} by essentially 1-based types by mean of Υ.

Proof: Υ will be fixed and we’ll freely omit the phrase ”by mean of Υ”. To see the ”In particular” part, work over A and assume that every p′ ∈ S(A′), with A′ ⊇ A countable, that is internal in p_{0}, is essentially 1-based over A. Indeed, assume p ∈ S(A) is such that every non-algebraic extension of p is non-foreign to p_{0}. Then, for a ⊧ p there exists a′ ∈ dcl^{eq}(Aa)\acl^{eq}(A) such that tp(a′/A) is p_{0}-internal and thus essentially 1-based over A by our assumption. Since L and Aa are countable so is dcl^{eq}(Aa) and thus by repeating this process we get that p is analyzable in p_{0} by essentially 1-based types. We prove now the main part. Assume there exist a countable A and p ∈ S(A) that is internal in p_{0} and p is not essentially 1-based. By Fact 2.2, we may assume there exists d ⊧ p, and b that is independent from d over A, and a finite tuple c̄ ⊆ p_{0} such that d ∈ dcl(Abc̄), and there exists a type-definable Υ-open set U over Ad such that a is independent from A for all a ∈ U and {a ∈ U|Cb(a/Ad) ⊈ acl^{eq}(a)} is not nowhere dense in the Stone-topology of U. So, since Υ refines the Stone-topology, by intersecting it with a definable set, we may assume that {a ∈ U|Cb(a/Ad) ⊈ acl^{eq}(a)} is dense in the Stone-topology of U. Now, for each disjoint partition c̄ = c̄_{0}c̄_{1} and formula χ(x̄_{1}, x̄_{0}, y, z) ∈ L(A) such that (*) ∀x̄_{0}, y, z∃^{<∞}x̄_{1}χ(x̄_{1}, x̄_{0}, y, z), let

F_{χ,c̄_{0},c̄_{1}} = {a ∈ U| ∃b′, c̄′_{0}, c̄′_{1} s.t. tp(b′c̄′_{0}c̄′_{1}/Ad) = tp(bc̄_{0}c̄_{1}/Ad) and a is independent from

b′c̄′_{0}c̄′_{1} over Ad and ⊧ χ(c̄′_{1}, c̄′_{0}, b′, a) and a is independent from Ab′c̄′_{0} over ∅}.

Let Pc̄ be the (finite) set of partitions of c̄ into two subsets. Note that since d is independent from b over A, any a ∈ U is independent from Ab′ whenever

tp(b′/Ad) = tp(b/Ad) and \begin{aligned}a & ⌣ | & b′ \\ & Ad \end{aligned} . Thus, since p_{0} is a partial type over

∅ of SU-rank ≤ 1 we have

U = ⋃_{(c̄_{0},c̄_{1})∈Pc̄, χ⊧(∗)} F_{χ,c̄_{0},c̄_{1}}.

Note that since we are fixing the type of b′c̄′_{0}c̄′_{1} over Ad, the sets F_{χ,c̄_{0},c̄_{1}} are type-definable over Ad. Since L and A are countable, by the Baire category theorem for the Stone-topology of the closed set U, there exists (c̄_{0}^{∗}¯,c_{1}^{∗}) ∈ Pc̄ and there is χ^{∗} ⊧ (∗) such that F_{χ^{∗},c̄_{0}^{∗},c̄_{1}^{∗}} has non-empty interior in the Stonetopology of U. Thus, we may assume that U is a type-definable Υ-open set over Ad such that {a ∈ U|Cb(a/Ad) ⊈ acl^{eq}(a)} is dense in the Stonetopology of U and for every a ∈ U there exists b′c̄′_{0}c̄′_{1} ⊧ tp(bc̄_{0}^{∗}c̄_{1}^{∗}/Ad) that is independent from a over Ad and such that ⊧ χ^{∗}(c̄′_{1}, c̄′_{0}, b′, a) and a is independent fromAb′c̄′_{0} over ∅. Let us now define a set V over Ad by

V = {(c̄′_{0}, c̄′_{1}, b′, a′, e′)| if tp(b′c̄′_{0}c̄′_{1}/Ad) = tp(bc̄_{0}^{∗}c̄_{1}^{∗}/Ad) and a′ is independent from

b′c̄′_{0}c̄′_{1} over Ad and a′ is independent from Ab′c̄′_{0} over ∅ and ⊧ χ^{∗}(c̄′_{1}, c̄′_{0}, b′, a′)

then e′ ∈ acl(Cb(Ab′c̄′_{0}c̄′_{1}/a′))}.

Let

V^{∗} = {e′|∃a′ ∈ U ∀b′, c̄′_{0}, c̄′_{1} V (c̄′_{0}, c̄′_{1}, b′, a′, e′)}.

Subclaim 5.7 V^{∗} is a Υ-open set over Ad.

Proof: By Proposition 4.4, we see that V is a Stone-open set over Ad. Since Stone-open sets are closed under the ∀ quantifier and the Υ topology refines the Stone-topology and closed under product by a full Stone-space and closed under projections, we conclude that V^{∗} is a Υ-open set.

Subclaim 5.8 For appropriate sort for e′, the set V^{∗} is unbounded and has finite SU-rank over Ad.

Proof: Let a^{∗} ∈ U be such that Cb(a^{∗}/Ad) ⊈ acl^{eq}(a^{∗}). Then Cb(Ad/a^{∗}) ⊈ acl^{eq}(Ad). By Remark 5.5, there exists e^{∗} ∉ acl^{eq}(Ad) such that e^{∗} ∈ acl^{eq}(Cb(Ab′c̄′_{0}c̄′_{1}/a^{∗})) for all b′c̄′_{0}c̄′_{1} ⊧ tp(bc̄_{0}^{∗}c̄_{1}^{∗}/Ad). In particular, e^{∗} ∈ V^{∗}. Thus, if we fix the sort for e′ in the definition of V^{∗} to be the sort of e^{∗}, then V^{∗} is unbounded. Now, let e′ ∈ V^{∗}. Then for some a′ ∈ U, ⊧ V (c̄′_{0}, c̄′_{1}, b′, a′, e′) for all b′, c̄′_{0}, c̄′_{1}. By what we saw above, there exists b′c̄′_{0}c̄′_{1} ⊧ tp(bc̄_{0}^{∗}c̄_{1}^{∗}/Ad) that is independent from a′ over Ad such that ⊧ χ^{∗}(c̄′_{1}, c̄′_{0}, b′, a′) and a′ is independent from Ab′c̄′_{0} over ∅. Thus, by the definition of V^{∗}, e′ ∈ acl(Cb(Ab′c̄′_{0}c̄′_{1}/a′)). Since Ab′ is independent from a′ over ∅, tp(e′) is almost-p_{0}-internal, and thus SU(e′) < ω. In particular, SU(e′/Ad) < ω.

Thus V^{∗} is the required set.

6 Stable dependence

The goal of this section is to prove the symmetry of the relation stabledependence. In this section T is assumed to be a complete theory unless otherwise stated, and we work in C^{eq}.

Definition 6.1 Let a ∈ C, A ⊆ B ⊆ C. We say that a is stably-independent from B over A if for every stable ϕ(x, y) ∈ L, if ϕ(x, b) is over B (i.e. the canonical parameter of ϕ(x, b) is in dcl(B)) and a′ ⊧ ϕ(x, b) for some a′ ∈ dcl(Aa), then ϕ(x, b) doesn’t divide over A. In this case we denote it by

\begin{aligned}a & ⌣ |s & B \\ & A \end{aligned} .

We will need some basic facts from local stability [HP]. From now on we fix a stable formula ϕ(x, y). A formula ψ ∈ L(C) is said to be a ϕ-formula over A if it is a finite boolean combination of instances of ϕ, that is equivalent to a formula with parameters from A. A complete ϕ-type over A is a consistent complete set of ϕ-formulas over A. S_{ϕ}(A) denotes the set of complete ϕ-types over A. Note that if M is a model then every p ∈ S_{ϕ}(M) is determined by the set {ψ ∈ p| ψ = ϕ(x, a) or ψ = ¬ϕ(x, a) for a ∈ M} (in fact, it is easy to see that every ϕ-formula over M is equivalent to a ϕ-formula whose parameters are from M). Recall the following well known facts.

Fact 6.2 Let ϕ(x, y) ∈ L be stable. Then

1) For any model M, every p ∈ S_{ϕ}(M) is definable.

2) Let A be any set, let p ∈ S(A), and let M ⊇ A be a model. Then there exists q ∈ S_{ϕ}(M) that is consistent with p and is definable over acl^{eq}(A).

3) Let A = acl(A). Let p ∈ S_{ϕ}(A). Then for every model M ⊇ A, there is a unique p̄ ∈ S_{ϕ}(M) that extends p and such that p̄ is definable over A (i.e. its ϕ-definition is over A). Moreover, there is a canonical formula over A that is the definition of any such p̄ over any such model M.

4) Assume p, q ∈ S_{ϕ}(acl(A)) are such that p|A = q|A. Then there exists σ ∈ Aut(C/A) such that σ(p) = q.

The following definition is standard.

Definition 6.3 Let p ∈ S_{ϕ}(B) and let A ⊆ B. We say that p doesn’t fork over A in the sense of local stability (=LS) if for some model M containing B and some p̄ ∈ S_{ϕ}(M) that extends p, p̄ is definable over acl(A).

The following lemma is easy but important.

Lemma 6.4 Assume T is a simple theory in which Lstp=stp over sets and let ϕ(x, y) ∈ L be stable. Then for all a and A ⊆ B ⊆ C, tp_{ϕ}(a/B) doesn’t fork over A in the sense of LS iff tp_{ϕ}(a/B) doesn’t fork over A.

Proof: Assume p_{ϕ} = tp_{ϕ}(a/B) doesn’t fork over A in the sense of LS. Extend it to a complete ϕ-type p̄_{ϕ} over a sufficiently saturated and sufficiently strongly-homegeneous model M that is definable over acl(A). If tp_{ϕ}(a/B) divide over A, there is an acl(A)-indiscernible sequence (B_{i}|i < ω) ⊆ M such that if p_{B_{i}}^{ϕ} are the corresponding acl(A)-conjugates of p_{ϕ}, then ⋀_{i} p_{B_{i}}^{ϕ} = ∅. By the uniqueness of non-forking extensions (in the sense of LS) of complete ϕ-types over algebraically closed sets (and the fact that M is sufficiently strongly-homegeneous) we conclude that p̄_{ϕ} extends each p_{B_{i}}^{ϕ}, a contradiction. For the other direction, assume p_{ϕ} = tp_{ϕ}(a/B) doesn’t fork over A. Let M ⊇ B be a sufficiently saturated and sufficiently strongly homogeneous model. Let p̄ ∈ S(M) be an extension of p_{ϕ} that doesn’t fork over A. Let ψ(x, c) ∈ L(M) be the definition of p̄|ϕ (where c is the canonical parameter of ψ). We claim that c ∈ acl(A). Indeed, otherwise let σ ∈ Aut(M/acl(A)) be such that σc ≠ c. So, p̄, σ(p̄) have different ϕ-definitions, contradiction to the following claim:

Claim 6.5 Let T be simple. Let ϕ(x, y) ∈ L be stable. Assume \begin{aligned}a & ⌣ | & b \\ & A \end{aligned}

and \begin{aligned}a′ & ⌣ | & b \\ & A \end{aligned} and Lstp(a/A) = Lstp(a′/A). Then ϕ(a, b) iff ϕ(a′, b).

Corollary 6.6 Let T be a simple theory in which Lstp=stp over sets. Then

for all a, A ⊆ B ⊆ C we have \begin{aligned}a & ⌣ |s & B \\ & A \end{aligned} iff tp_{ϕ}(a′/B) doesn’t fork over A in

the sense of LS for every stable ϕ(x, y) ∈ L and every a′ ∈ dcl(aA).

Given a, A ⊆ B ⊆ C, we will say that tp(a/B) doesn’t fork over A in the sense of LS if the right hand side of Corollary 6.6 holds.

Lemma 6.7 Let T be a simple theory in which Lstp=stp over sets. Then 1) stable independence is a symmetric relation, that is, for all a, b, A we have

\begin{aligned}a & ⌣ |s & Ab \\ & A \end{aligned} iff \begin{aligned}b & ⌣ |s & Aa \\ & A \end{aligned} .

2) For all a, A ⊆ B ⊆ C, if \begin{aligned}a & ⌣ |s & B \\ & A \end{aligned} and \begin{aligned}a & ⌣ |s & C \\ & B \end{aligned} , then \begin{aligned}a & ⌣ |s & C \\ & A \end{aligned} . In

fact, in any theory the same is true in the sense of LS.

Proof: To prove 1), first note the following.

Subclaim 6.8 Let ϕ(x, y) ∈ L be stable and let a, a′ ∈ C and let A ⊆ C. Assume tp_{ϕ}(a/A) = tp_{ϕ}(a′/A). Then ϕ(a, y) forks over A iff ϕ(a′, y) forks over A.

Proof: Otherwise, there are p, q ∈ S(C), both extends tp_{ϕ}(a/A) = tp_{ϕ}(a′/A), and do not fork over A such that p represent ϕ(x, y) (namely, for some b ∈ M, ϕ(x, b) ∈ p) and q doesn’t represent ϕ(x, y). By Fact 6.2 (4), (p|ϕ)|acl(A) and (q|ϕ)|acl(A) are A-conjugate. Let σ ∈ Aut(C/A) be such that σ((p|ϕ)|acl(A)) = (q|ϕ)|acl(A). Now, both σ(p|ϕ) and q|ϕ extend (q|ϕ)|acl(A) and doesn’t fork over acl(A), and therefore by Lemma 6.4, both doesn’t fork over acl(A) in the sense of LS. By Fact 6.2 (3), σ(p|ϕ) = q|ϕ, which is a contradiction.

We prove symmetry. Assume \begin{aligned}a & ⌣ |s & Ab \\ & A \end{aligned} . To show \begin{aligned}b & ⌣ |s & Aa \\ & A \end{aligned} , let ϕ(x, y) ∈

L be stable such that ϕ(b′, a′) for some b′ ∈ dcl(Ab) and some a′ ∈ dcl(Aa). By the assumption, tpϕ̃(a′/Ab) doesn’t fork over A (in the usual sense), so

there exists a′′ ⊧ tpϕ̃(a′/Ab) such that \begin{aligned}a′′ & ⌣ | & Ab \\ & A \end{aligned} . Let (a′′_{i}|i < ω) be

a Morley sequence of tp(a′′/Ab). Now, b′ ⊧ ⋀_{i<ω} ϕ(x, a′′_{i}). Thus ϕ(x, a′′) doesn’t fork over A. By Subclaim 6.8, ϕ(x, a′) doesn’t fork over A. 2) is immediate by Corollary 6.6 and fact that the relation of being a non-forking extension in the LS sense is a transitive relation on complete ϕ-types (where ϕ is a fixed stable formula).

7 An unbounded τ_{∞}^{f}-open set of bounded finite SU_{s}-rank is sufficient

In this section T is an imaginary simple theory. We work in C^{eq}.

Definition 7.1 For a ∈ C and A ⊆ C the SU_{s}-rank is defined by induction

on α by SU_{s}(a/A) ≥ α + 1 if there exists B ⊇ A such that \begin{aligned}a & ⌣̸ |s & B \\ & A \end{aligned} and

SU_{s}(a/B) ≥ α.

Definition 7.2 The τ_{∞}^{f}-topology on S(A) is the topology whose basis is the family of type-definable τ^{f}-open sets over A.

Lemma 7.3 For a ∈ C and A ⊆ B ⊆ C, assume tp(a/B) doesn’t fork over

acl(aA) ∩ acl(B) and \begin{aligned}a & ⌣̸ | & B \\ & A \end{aligned} . Then \begin{aligned}a & ⌣̸ |s & B \\ & A \end{aligned} .

Proof: It will be sufficient to show that whenever \begin{aligned}a & ⌣̸ | & B \\ & A \end{aligned} and \begin{aligned}a & ⌣ | & B \\ & acl(a) ∩ acl(B) \end{aligned}

for some (possibly infinite) tuple a and some A ⊆ B, there exists a stable ϕ(x, y) ∈ L such that ϕ(a, B) and ϕ(x, b) forks over A (indeed, the above

implies the following: if \begin{aligned}aA & ⌣̸ | & B \\ & A \end{aligned} and \begin{aligned}aA & ⌣ | & B \\ & acl(aA) ∩ acl(B) \end{aligned} then

there exists a stable formula ϕ(x, y) ∈ L such that ϕ(aA, B) and ϕ(x, B)

forks over A, i.e. \begin{matrix}a & ⌣̸ |s & B \\ & A \end{matrix} ). To prove this, let E = Cb(a/B). Then

E ⊆ acl(a) ∩ acl(B). By the assumption, there is e^{∗} ∈ dcl(E)\acl(A), so e^{∗} ∈ (acl(a) ∩ acl(B))\acl(A). Hence there are n_{0}, n_{1} ∈ ω and formulas χ_{0}(x, y), χ_{1}(x, z) ∈ L such that ∀y∃^{<n_{0}}xχ_{0}(x, y) and ∀z∃^{<n_{1}}xχ_{0}(x, z) and χ_{0}(e^{∗}, a) and χ_{1}(x, B) isolates tp(e^{∗}/B). Let

ϕ(y, z) ≡ ∃x(χ_{0}(x, y) ∧ χ_{1}(x, z)).

Note that ϕ(y, z) is stable. Indeed, otherwise there are a ∈ C and an ∅-indiscernible sequence B = (b_{i}|i ∈ Z) (Z=the integer numbers) such that i ≥ 0 iff ϕ(a, b_{i}). Since B is indiscernible, and χ_{1}(x, b_{0}) is algebraic, ⋂_{i∈I} χ_{1}(C, b_{i}) = ⋂_{i∈Z} χ_{1}(C, b_{i}) for every infinite I ⊆ Z. But since χ_{0}(x, a) is algebraic, for some infinite I^{∗} ⊆ ω, χ_{0}(C, a) ∩ ⋂_{i∈I^{∗}} χ_{1}(C, b_{i}) ≠ ∅. A contradiction to ¬ϕ(a, b_{i}) for i < 0. To see that ϕ(y, B) forks over A,

note that otherwise there exists a′ ⊧ ϕ(y, B) such that \begin{aligned}a′ & ⌣ | & B \\ & A \end{aligned} , so if

e′ ⊧ χ_{0}(x, a′) ∧ χ_{1}(x, B), then on one hand \begin{aligned}e′ & ⌣ | & B \\ & A \end{aligned} and on the other hand since χ_{1}(x, B) isolates tp(e^{∗}/B), e′ ∈ acl(B)\acl(A) which is a contradiction.

Lemma 7.4 Assume U is an unbounded τ_{∞}^{f}-open set of bounded finite SU_{s}- rank over some finite set A. Then there exists a τ_{∞}^{f}-open set U^{∗} ⊆ U over some finite set B^{∗} ⊇ A of SU_{s}-rank 1.

Proof: We may clearly assume U is a basic τ_{∞}^{f}-open set. Let n = SU_{s}(U) (U is over A, and n < ω). Let a^{∗} ∈ U with SU_{s}(a^{∗}/A) = n. Let B ⊇ A be finite

such that \begin{aligned}a^{∗} & ⌣̸ |s & B \\ & A \end{aligned} , and SU_{s}(a^{∗}/B) = n−1. So, there exists a′ ∈ dcl(a^{∗}A)

and stable ϕ(x, y) ∈ L such that ϕ(a′, B) and ϕ(x, B) forks over A. Let f an ∅-definable function such that a′ = f(a^{∗}, A). Let

U′ = {a ∈ U| ϕ(f(a, A), B) } (as a set over B).

Since a^{∗} ∈ U′, SU_{s}(U′) ≥ n − 1. If a ∈ U′, then ϕ(f(a, A), B) implies

\begin{aligned}a & ⌣̸ |s & B \\ & A \end{aligned} and therefore SU_{s}(U′) ≤ n − 1. We conclude SU_{s}(U′) = n − 1.

Clearly, U′ ⊆ U and U′ is type-definable. By Fact 2.7, U′, is a τ^{f}-open set over B. We finish by induction.

Lemma 7.5 Let T be a countable imaginary simple unidimensional theory. Assume there is p_{0} ∈ S(∅) of SU-rank 1 and there exists an unbounded τ_{∞}^{f}-open set over some finite set of bounded finite SU_{s}-rank. Then T is supersimple.

Proof: By Lemma 7.4, there exists a finite set A_{0} and a τ_{∞}^{f}-open set U over A_{0} of SU_{s}-rank 1. Clearly, we may assume U is type-definable. By Theorem 3.12, T is PCFT. Thus, working over A_{0}, by Theorem 5.6 for the τ^{f}-topology either (i) there exists an unbounded τ^{f}-open set of finite SU-rank over some countable set or (ii) every non-algebraic type over A_{0} is analyzable in p_{0} by essentially 1-based types by mean of τ^{f}. By Corollary 3.13, we may assume (ii). We claim SU(U) = 1. Indeed, otherwise there exists a and d ∈ U

such that \begin{aligned}d & ⌣̸ | & a \\ & A_{0} \end{aligned} and d ∉ acl(aA_{0}). By (ii), there exists (a_{i}|i ≤ α) ⊆

dcl^{eq}(aA_{0}) with a_{α} = a such that tp(a_{i}/A_{0} ∪ {a_{j}|j < i}) is essentially 1-based over A_{0} for all i ≤ α. Now, let i^{∗} ≤ α be minimal such that there exists

d′ ∈ U satisfying \begin{aligned}d′ & ⌣̸ | & \{a_{i}|i ≤ i^{∗}\} \\ & A_{0} \end{aligned} , and d′ ∉ acl(A_{0} ∪ \{a_{i}|i ≤ i^{∗}\}). Pick

ϕ(x, a′) ∈ L(A_{0} ∪ {a_{i}|i ≤ i^{∗}}) that forks over A_{0} and such that ϕ(d′, a′). Let

V = {d ∈ U| ϕ(d, a′) and d ∉ acl(A_{0} ∪ {a_{i}|i ≤ i^{∗}}) }.

By minimality of i^{∗}, d is independent from {a_{i}|i < i^{∗}} over A_{0} for all d ∈ V. Clearly V is type-definable and by Fact 2.7, V is a τ^{f}-open set over A_{0} ∪ {a_{i}|i ≤ i^{∗}}. Now, since tp(a_{i^{∗}}/A_{0} ∪ {a_{i}|i < i^{∗}}) is essentially 1-based over A_{0}, the set

{d ∈ V | Cb(d/A_{0} ∪ {a_{i}|i ≤ i^{∗}}) ∈ bdd(dA_{0})}

contains a relatively Stone-dense and open subset of V. In particular, there exists d^{∗} ∈ V such that tp(d^{∗}/A_{0} ∪ {a_{i}|i ≤ i^{∗}} doesn’t fork over acl(A_{0}d^{∗}) ∩ acl(A_{0} ∪ {a_{i}|i ≤ i^{∗}}).

Since we know \begin{aligned}d^{∗} & ⌣̸ | & A_{0} ∪ \{a_{i}|i ≤ i^{∗}\} \\ & A_{0} \end{aligned} , Lemma 7.3 implies \begin{aligned}d^{∗} & ⌣̸ |s & A_{0} ∪ \{a_{i}|i ≤ i^{∗}\} \\ & A_{0} \end{aligned} .

Hence d^{∗} ∈ V implies SU_{s}(d^{∗}/A_{0}) ≥ 2, which contradict SU_{s}(U) = 1. Thus we have proved SU(U) = 1. Now, by Fact 2.8 there exists a definable set of SU-rank 1, and thus by Fact 2.4, T is supersimple.

Remark 7.6 Note that if X is any Stone-closed subset of the Stone-space S_{x}(T) and B = {F_{i}}_{i∈I} is a basis for a topology τ on X that consists of Stone-closed subsets of X, then (X, τ) is a Baire space (i.e. the intersection countably many dense open sets in it is dense). In particular, the τ_{∞}^{f}-topology on S(A) is Baire.

Remark 7.7 If we could show that for all a, A ⊆ B ⊆ C,

\begin{aligned}a & ⌣ |s & C \\ & A \end{aligned} ⇒ \begin{aligned}a & ⌣ |s & C \\ & B \end{aligned} ,

then this would imply that for A ⊆ B, \begin{aligned}a & ⌣ |s & B \\ & A \end{aligned} implies SU_{s}(a/A) =

SU_{s}(a/B). Thus by Remark 7.6, a Baire categoricity argument, applying Theorem 3.12, will imply the existence of a bounded finite SU_{s}-rank unbounded τ_{∞}^{f}-open set in any countable imaginary unidimensional simple theory and thus supersimplicity will follow by Lemma 7.5. Unfortunately, this seems to be false for a general simple theory without stable forking.

8 τ̃^{f} and τ̃_{st}^{f}-sets

In this section T is assumed to be a simple theory. We work in C.

Definition 8.1 A relation V (x, z_{1}, ...z_{l}) is said to be a pre-τ̃^{f}-set relation if there are θ(x, x̃, z_{1}, z_{2}, ..., z_{l}) ∈ L and ϕ_{i}(x̃, y_{i}) ∈ L for 0 ≤ i ≤ l such that for all a, d_{1}, ..., d_{l} ∈ C we have

V (a, d_{1}, ..., d_{l}) iff ∃ã [θ(a, ã, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}...d_{i})]

(for i = 0 the sequence d_{1}d_{2}...d_{i} is interpreted as ∅). If each ϕ_{i}(x̃, y_{i}) is assumed to be stable, V (x, z_{1}, ...z_{l}) is said to be a pre-τ̃_{st}^{f}-set relation.

Definition 8.2 1) A τ̃^{f}-set (over ∅) is a set of the form

U = {a| ∃d_{1}, d_{2}, ...d_{l} V (a, d_{1}, ..., d_{l})}

for some pre-τ̃^{f}-set relation V (x, z_{1}, ...z_{l}). The minimal natural number l for which the set U above can be defined is called the degree of the set U (e.g. τ^{f}-open sets have degree 0).

2) A τ̃_{st}^{f}-set is defined in the same way as a τ̃^{f}-set but we add the requirement that V (x, z_{1}, ...z_{l}) is a pre-τ̃_{st}^{f}-set relation.

We will say that the formula ϕ(x, y) ∈ L is low in x if there exists k < ω such that for every ∅-indiscernible sequence (b_{i}|i < ω), the set {ϕ(x, b_{i})|i < ω} is inconsistent iff every subset of it of size k is inconsistent. Note that every stable formula ϕ(x, y) is low in both x and y.

Remark 8.3 Note that if ϕ(x, y) ∈ L is low in x then the relation F_{ϕ} defined by F_{ϕ}(b, A) iff ϕ(x, b) forks over A is type-definable. Thus every pre-τ̃_{st}^{f}-set relation is type-definable and every τ̃_{st}^{f}-set is type-definable.

Lemma 8.4 Assume the extension property is first-order in T. Let θ(x, z_{1}, ..., z_{n}) be a Stone-open relation over ∅ and let ϕ_{j}(x, y_{j}) ∈ L for j = 0, .., n. Let U be the following invariant set. For all d_{1} ∈ C, U(d_{1}) iff

∃a, d_{2}, d_{3}, ...d_{n}[θ(a, d_{1}, ...d_{n}) ∧ ⋀_{j=0}^{n} ϕ_{j}(a, y_{j}) forks over d_{1}...d_{j}].

Then U is a τ^{f}-open set over ∅. If each ϕ_{j}(x, y_{j}) is assumed to be low in y_{j} and θ is assumed to be definable, then U is a basic τ_{∞}^{f}-open set.

Proof: We prove the lemma by induction on n ≥ 1. Consider the negation Γ of U:

Γ(d_{1}) iff ∀a∀d_{2}...d_{n}(θ(a, d_{1}, ...d_{n}) → ⋁_{j=0}^{n} ϕ_{j}(a, y_{j}) dnfo d_{1}...d_{j})

(where ”dnfo”=doesn’t fork over).

Subclaim 8.5 Let Γ′ be defined by Γ′(d_{1}) iff

⋀_{{η_{j}}_{j=0}^{n−1}∈L} ∀d_{2}...d_{n}[( ⋀_{j=0}^{n−1} η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}...d_{j}) → ∀aΛ(a, d_{1}, ..., d_{n})].

where Λ is defined by

Λ(a, d_{1}, ...d_{n}) iff θ(a, d_{1}, ...d_{n}) → ⋁_{j=0}^{n} ϕ_{j}(a, y_{j}) ∧ ¬η_{j}(d_{1}...d_{n}, y_{j}) dnfo d_{1}...d_{n}

where η_{n} denotes a contradiction. Then Γ′ = Γ.

Proof: Assume Γ(d_{1}). Let η_{0}, ...n_{n−1} ∈ L and let d_{2}, ...d_{n} ∈ C. Assume η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}...d_{j} for all 0 ≤ j ≤ n − 1, and let a ∈ C be such that θ(a, d_{1}, ...d_{n}). By the assumption, ϕ_{j_{0}}(a, y_{j_{0}}) doesn’t fork over d_{1}...d_{j_{0}} for some 0 ≤ j_{0} ≤ n − 1. Let c_{j_{0}} be such that ϕ_{j_{0}}(a, c_{j_{0}})

and \begin{aligned}a & ⌣ | & c_{j_{0}} \\ & d_{1}...d_{j_{0}} \end{aligned} . By extension we may assume \begin{aligned}ad_{1}...d_{n} & ⌣ | & c_{j_{0}} \\ & d_{1}...d_{j_{0}} \end{aligned} .

Since η_{j_{0}}(d_{1}...d_{n}, y_{j_{0}}) forks over d_{1}...d_{j_{0}}, we know ¬η_{j_{0}}(d_{1}...d_{n}, c_{j_{0}}). Therefore ϕ_{j_{0}}(a, y_{j_{0}}) ∧ ¬η_{j_{0}}(d_{1}...d_{n}, y_{j_{0}}) doesn’t fork over d_{1}...d_{j_{0}} and in particular doesn’t fork over d_{1}...d_{n}. Assume now Γ′(d_{1}). Let a, d_{2}, ...d_{n} ∈ C and assume θ(a, d_{1}, ...d_{n}). It is sufficient to show that for all 0 ≤ j ≤ n−1 if ϕ_{j}(a, y_{j}) forks over d_{1}...d_{j}, then there exists η_{j} such that η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}...d_{j} and ϕ_{j}(a, y_{j}) ∧ ¬η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}....d_{n}. Assume otherwise. Fix j, so ϕ_{j}(a, y_{j}) forks over d_{1}...d_{j} and ϕ_{j}(a, y_{j}) ∧ ¬η_{j}(d_{1}...d_{n}, y_{j}) doesn’t fork over d_{1}....d_{n} for all η_{j} such that η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}...d_{j}. Let

Ψ(y_{j}) ≡ ⋀_{η_{j}∈F_{j},μ_{j}∈E_{j}} ϕ_{j}(a, y_{j}) ∧ ¬η_{j}(d_{1}...d_{n}, y_{j}) ∧ ¬μ_{j}(ad_{1}...d_{n}, y_{j})

where

F_{j} = {η_{j}| η_{j}(d_{1}...d_{n}, y_{j}) forks over d_{1}...d_{j}},

and

E_{j} = {μ_{j}| μ_{j}(ad_{1}...d_{n}, y_{j}) forks over d_{1}...d_{n}}.

By our assumption and compactness, Ψ(y_{j}) is consistent. Let c_{j} ⊧ Ψ(y_{j}).

Then ϕ_{j}(a, c_{j}), \begin{aligned}d_{1}...d_{n} & ⌣ | & c_{j} \\ & d_{1}...d_{j} \end{aligned} , and \begin{aligned}ad_{1}...d_{n} & ⌣ | & c_{j} \\ & d_{1}...d_{n} \end{aligned} . By transi-

tivity, \begin{aligned}ad_{1}...d_{n} & ⌣ | & c_{j} \\ & d_{1}...d_{j} \end{aligned} . A contradiction to the assumption that ϕ_{j}(a, y_{j})

forks over d_{1}...d_{j}. The proof of Subclaim 8.5 is complete.

Since the extension property is first-order for T, the relation Λ_{0} defined by Λ_{0}(d_{1}, ...d_{n}) ≡ ∀aΛ(a, d_{1}, ...d_{n}) is type-definable. Now, clearly for all d_{1}, Γ′(d_{1}) iff

⋀_{{η_{j}}_{j=0}^{n−1}∈L} ∀d_{2}...d_{n}(¬Λ_{0}(d_{1}, ..., d_{n}) → ⋁_{j=0}^{n−1} η_{j}(d_{1}...d_{n}, y_{j}) dnfo d_{1}...d_{j}).

Now, if n = 1 then this is clearly τ^{f}-closed. If n > 1, then we finish by the induction hypothesis.

Corollary 8.6 Assume the extension property is first-order in T. Let m ≤ l < ω and let d_{1}^{∗}, ...d_{m}^{∗} ∈ C. Let θ ∈ L and ϕ_{i} ∈ L for i ≤ l. Let V be defined by

V (a, d_{1}, ..., d_{l}) iff [θ(a, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(a, y_{i}) forks over d_{1}d_{2}...d_{i})].

Then the set U defined by

U(d_{m+1}) iff ∃a∃d_{m+2}...d_{l} V (a, d_{1}^{∗}, ...d_{m}^{∗}, d_{m+1}, ...d_{l})

is a τ^{f}-open set over d_{1}^{∗}...d_{m}^{∗}.

Proof: By Fact 2.7, there are formulas {ψ_{j}(x̃, w_{j}) ∈ L(d_{1}^{∗}...d_{m}^{∗})}_{j∈J} such that

∀a [ ⋀_{i=0}^{m}(ϕ_{i}(a, y_{i}) forks over d_{1}^{∗}d_{2}^{∗}...d_{i}^{∗}) iff ⋁_{j∈J}(ψ_{j}(a, w_{j}) forks over d_{1}^{∗}d_{2}^{∗}...d_{m}^{∗})].

Therefore by Lemma 8.4 (since by Lemma 3.2, the extension property is first-order over d̄^{∗} as well) U is a union over j ∈ J of τ^{f}-open sets over d_{1}^{∗}d_{2}^{∗}...d_{m}^{∗}.

Theorem 8.7 Assume the extension property is first-order in T. Then

1) Let U be an unbounded τ̃^{f}-set over ∅. Then there exists an unbounded τ^{f}-open set U^{∗} over some finite set A^{∗} such that U^{∗} ⊆ U. In fact, if V (x, z_{1}, ..., z_{l}) is a pre-τ̃^{f}-set relation such that U = {a|∃d_{1}...d_{l}V (a, d_{1}, ..., d_{l})}, and (d_{1}^{∗}, ..., d_{m}^{∗}) is any maximal sequence (with respect to extension) such that ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l}) is unbounded, then

U^{∗} = ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l})

is a τ^{f}-open set over d_{1}^{∗}...d_{m}^{∗}.

2) Let U be an unbounded τ̃_{st}^{f}-set over ∅. Then there exists an unbounded τ_{∞}^{f}-open set U^{∗} over some finite set A^{∗} such that U^{∗} ⊆ U. In fact, if V (x, z_{1}, ..., z_{l}) is a pre-τ̃_{st}^{f}-set relation such that U = {a|∃d_{1}...d_{l}V (a, d_{1}, ..., d_{l})}, and (d_{1}^{∗}, ..., d_{m}^{∗}) is any maximal sequence (with respect to extension) such that ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l}) is non-algebraic, then

U^{∗} = ∃d_{m+1}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d_{m+1}, ..., d_{l})

is a basic τ_{∞}^{f}-open set over d_{1}^{∗}...d_{m}^{∗}.

Proof: By Remark 8.3, (2) is an immediate corollary of (1). It suffices, of course, to prove the second part of (1). T is PCFT by Corollary 3.5. Let d̄^{∗} = d_{1}^{∗}...d_{m}^{∗}. First, if m = l then the assertion follows immediately by Fact 2.7. So, we may assume m < l. By maximality of d̄^{∗}, we know ∃d_{m+2}...d_{l}V (C, d_{1}^{∗}, ..., d_{m}^{∗}, d′_{m+1}, d_{m+2}, ...d_{l}) is bounded (equivalently, a union of algebraic sets over d̄^{∗}) for every d′_{m+1}. Thus for every a ∈ U^{∗}, there exist χ_{a}(x, z̄^{∗}, z) ∈ L, k = k(χ_{a}) < ω and d′_{m+1}(a) ∈ C, such that ∀z∀z̄^{∗}∃^{=k}xχ_{a}(x, z̄^{∗}, z) (*1) and V (a, d_{1}^{∗}, ..., d_{m}^{∗}, d′_{m+1}(a), d_{m+2}, ...d_{l}) for some d_{m+2}, ...d_{l} ∈ C and χ_{a}(x, d̄^{∗}, d′_{m+1}(a)) isolates the type tp(a/d̄^{∗}, d′_{m+1}(a)). Let Ξ = {χ_{a}}_{a∈U^{∗}}. For χ ∈ Ξ, let k = k(χ) and let U_{χ} be the d̄^{∗}-invariant set defined by U_{χ}(d_{m+1}) iff

∃ distinct a_{1}....a_{k}[ ⋀_{j=1}^{k} χ(a_{j}, d̄^{∗}, d_{m+1})∧ ⋀_{j=1}^{k} ∃d_{m+2}...d_{l}V (a_{j}, d̄^{∗}, d_{m+1}, d_{m+2}, ...d_{l})]

Subclaim 8.8 U_{χ} is a τ^{f}-open set over d̄^{∗}.

Proof: Let V be given by:

V (a, d_{1}, ..., d_{l}) iff ∃ã [θ(a, ã, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}...d_{i})].

for some θ, ϕ_{i} ∈ L. Since T is PCFT, it is sufficient to show that there exists a τ^{f}-open set W = W (x, z_{m+1}, d̄^{∗}) over d̄^{∗} such that if U′_{χ} is defined by

U′_{χ}(d_{m+1}) iff ∃ distinct a_{1}....a_{k}[ ⋀_{j=1}^{k} χ(a_{j}, d̄^{∗}, d_{m+1}) ∧ ⋀_{j=1}^{k} W (a_{j}, d_{m+1}, d̄^{∗})]

then U′_{χ} = U_{χ}. To show this let W be defined by: W (a, d_{m+1}, d̄^{∗}) iff

∃ã∃d′_{m+2}...d′_{l}[θ(a, ã, d_{1}^{∗}, d_{2}^{∗}, ...d_{m}^{∗}, d_{m+1}, d′_{m+2}, ...d′_{l})∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, y_{i}) forks over d′_{1}d′_{2}...d′_{i})]

where d′_{i} is defined in the following way: for 1 ≤ i ≤ m, d′_{i} denotes d_{i}^{∗}, and d′_{m+1} denotes d_{m+1}a (and the rest are quantified variables). First note that for all a, d_{m+1} with a ∈ acl(d_{m+1}, d̄^{∗}), W (a, d_{m+1}, d̄^{∗}) iff ∃d_{m+2}...d_{l}V (a, d̄^{∗}, d_{m+1}, d_{m+2}, ...d_{l}). Thus by (*1), U′_{χ} = U_{χ}. By Corollary 8.6, W is a τ^{f}-open set over d̄^{∗}. So, the proof of Subclaim 8.8 is complete.

Now, for each χ ∈ Ξ define Y_{χ}(x) ≡ ∃d_{m+1}(χ(x, d̄^{∗}, d_{m+1}) ∧ U_{χ}(d_{m+1})). Since T is PCFT, Subclaim 8.8 implies Y_{χ} is a τ^{f}-open set over d̄^{∗}. Note that by the definition of U_{χ} and (*1), Y_{χ} ⊆ U^{∗} for all χ ∈ Ξ. Now, if a ∈ U^{∗}, then by the choice of d′_{m+1}(a), χ_{a} and k = k(χ_{a}), we have χ_{a}(a, d̄^{∗}, d′_{m+1}(a)) ∧ U_{χ_{a}}(d′_{m+1}(a)). Thus a ∈ Y_{χ_{a}}. Hence U^{∗} = ⋃_{χ∈Ξ} Y_{χ}, and so U^{∗} is a τ^{f}-open set over d̄^{∗}. The proof of Theorem 8.7 is complete.

9 Main Result

Definition 9.1 For a ∈ C and A ⊆ C the SU_{se}-rank is defined by induction

on α: SU_{se}(a/A) ≥ α + 1 if there exist B_{1} ⊇ B_{0} ⊇ A such that \begin{aligned}a & ⌣̸ |s & B_{1} \\ & B_{0} \end{aligned}

and SU_{se}(a/B_{1}) ≥ α.

Remark 9.2 Note that SU_{se}(a/B) ≤ SU_{se}(a/A) for all a ∈ C and A ⊆ B ⊆ C (this is the reason for introducing SU_{se}). Also, clearly SU_{s}(a/A) ≤ SU_{se}(a/A) ≤ SU(a/A) for all a, A. Clearly SU_{se}(a/A) = 0 iff SU_{s}(a/A) = 0 iff a ∈ acl(A) for all a, A.

Theorem 9.3 Let T be a countable imaginary simple unidimensional theory. Then T is supersimple.

Proof: By adding countably many constants we may assume there exists p_{0} ∈ S(∅) of SU-rank 1 (each of the assumptions is preserved, as well as the corollary). By Lemma 7.5, it will be sufficient to show there exists an unbounded τ_{∞}^{f}-open set of bounded finite SU_{s}-rank over some finite set. Fix a non-algebraic sort s. Since T is unidimensional and imaginary, by Fact 2.1 for every a ∈ C^{s}\acl(∅) there exists a′ ∈ dcl(a)\acl(∅) such that tp(a′) is p_{0}- internal and thus has finite SU-rank. To finish the proof it will be sufficient to show that in any countable simple theory T in which the extension property is first-order, the non-existence of an unbounded τ_{∞}^{f}-open set of bounded finite SU_{se}-rank over a finite set implies ∃a^{∗} ∈ C^{s}\acl(∅) such that for every ∅-definable function f, either f(a^{∗}) ∈ acl(∅) or SU_{se}(f(a^{∗})) ≥ ω. To show this, assume the above assumptions on T. For every ∅-definable function f and n < ω, let

S_{f,n} = {a ∈ C^{s}| 0 < SU_{se}(f(a)) < n}.

Subclaim 9.4 For every non-empty τ̃_{st}^{f}-set U ⊆ C^{s} (with U ∩ acl(∅) = ∅) for all ∅-definable function f, and n < ω, there exists a non-empty τ̃_{st}^{f}-set U^{∗} ⊆ U ∩ (C^{s}\S_{f,n}).

Assuming Subclaim 9.4 is true, let ((f_{i}, n_{i})|i < ω) be an enumeration of all such pairs (f, n). By induction, let U_{0} = C^{s}\acl(∅), and let U_{i+1} ⊆ U_{i} ∩ (C\S_{f_{i},n_{i}}) be a non-empty τ̃_{st}^{f}-set. Since each U_{i} is type-definable, by compactness ⋂_{i<ω} U_{i} ≠ ∅. So, any a^{∗} ∈ ⋂_{i<ω} U_{i} will work.

Proof of Subclaim 9.4: Let U, (f, n) be as in Subclaim 9.4. Now, if there exists a ∈ U such that f(a) ∈ acl(∅), let χ(x) ∈ L be algebraic such that χ(f(a)). By letting U^{∗} = {a ∈ U| ⊧ χ(f(a))} we are done. Hence we may assume f(a) ∉ acl(∅) for every a ∈ U. Let V (x, z_{1}, ...z_{n}) be a pre-τ̃_{st}^{f}-set relation such that

U = {a| ∃d_{1}, d_{2}, ...d_{l} V (a, d_{1}, ..., d_{l})}.

where V is defined by:

V (a, d_{1}, ..., d_{l}) iff ∃ã [θ(a, ã, d_{1}, d_{2}, ..., d_{l}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}...d_{i})]

for some θ(x, x̃, z_{1}, z_{2}, ..., z_{l}) ∈ L and stable ϕ_{i}(x̃, y_{i}) ∈ L for 0 ≤ i ≤ l. Now, let V_{f} be defined by: for all b, d_{1}, ..., d_{l} ∈ C,

V_{f}(b, d_{1}, ..., d_{l}) iff ∃a(b = f(a) ∧ V (a, d_{1}, ..., d_{l})).

Then, clearly V_{f} is a pre-τ̃_{st}^{f}-set relation. Let

U_{f} = {b| ∃d_{1}, d_{2}, ...d_{l} V_{f}(b, d_{1}, ..., d_{l})}.

Let d̄^{∗} = (d_{1}^{∗}, ..., d_{m}^{∗}) be a maximal sequence, with respect to extension, (m ≤ l) such that

Ṽ_{f}(v) ≡ ∃d_{m+1}, d_{m+2}, ...d_{l}V_{f}(v, d_{1}^{∗}, ...d_{m}^{∗}, d_{m+1}, ...d_{l})

is non-algebraic, or equivalently unbounded (since U ≠ ∅ and we assume f(a) ∉ acl(∅) for all a ∈ U, the empty sequence satisfies this property). By Theorem 8.7, Ṽ_{f}(C) is a basic τ_{∞}^{f}-open set over d̄^{∗}. By our assumption Ṽ_{f}(C) is not of bounded finite SU_{se}-rank. Thus there are a^{∗} and d_{m+1}^{∗}, ...d_{l}^{∗} such that V (a^{∗}, d̄^{∗}, d_{m+1}^{∗}, ..., d_{l}^{∗}) and SU_{se}(f(a^{∗})/d̄^{∗}) ≥ n. Let E = ⟨(c_{i}^{∗}, e_{i}^{∗})|1 ≤ i ≤ n⟩

be such that \begin{aligned}f(a^{∗}) & ⌣̸ |s & e_{i}^{∗} \\ & d̄^{∗}c_{1}^{∗}e_{1}^{∗}...c_{i}^{∗} \end{aligned} for all 1 ≤ i ≤ n (*1). Note that since

both dcl and forking have finite character, we may assume that c_{i}^{∗}, e_{i}^{∗} are finite tuples. Let ã^{∗} be such that:

θ(a^{∗}, ã^{∗}, d_{1}^{∗}, d_{2}^{∗}, ..., d_{l}^{∗}) ∧ ⋀_{i=0}^{l}(ϕ_{i}(ã^{∗}, y_{i}) forks over d_{1}^{∗}d_{2}^{∗}...d_{i}^{∗}) (∗2).

Now, by maximality of d̄^{∗}, f(a^{∗}) ∈ acl(d̄^{∗}d_{m+1}^{∗}). By taking a non-forking extension of tp(E/acl(d̄^{∗}d_{m+1}^{∗})) over acl(d_{1}^{∗}...d_{l}^{∗}a^{∗}ã^{∗}) we may assume that

\begin{aligned}a^{∗}ã^{∗}d_{1}^{∗}...d_{l}^{∗} & ⌣ | & E \\ & d̄^{∗}d_{m+1}^{∗} \end{aligned} and (*1) and (*2) still hold. Thus \begin{aligned}a^{∗}ã^{∗} & ⌣ | & d_{1}^{∗}...d_{i}^{∗}E \\ & d_{1}^{∗}...d_{i}^{∗} \end{aligned}

for all m+1 ≤ i ≤ l. Hence by (*2), we conclude ϕ_{i}(ã^{∗}, y_{i}) forks over d_{1}^{∗}d_{2}^{∗}...d_{i}^{∗}E

for all m+1 ≤ i ≤ l. By (*1) and symmetry of ^{⌣ |s} (Lemma 6.7), there are stable ψ_{i}(x_{i}, w_{i}) ∈ L and ∅-definable functions g_{i} for 1 ≤ i ≤ n such that if a_{i}^{∗} = g_{i}(f(a^{∗}), d̄^{∗}c_{1}^{∗}e_{1}^{∗}...c_{i}^{∗}), then ψ_{i}(a_{i}^{∗}, e_{i}^{∗}) and ψ_{i}(a_{i}^{∗}, w_{i}) forks over d̄^{∗}c_{1}^{∗}e_{1}^{∗}...c_{i}^{∗}. Now, let us define a relation V^{∗} in the following way:

V^{∗}(a, d_{1}, ...d_{m}, c_{1}, e_{1}, ...c_{n}, e_{n}, d_{m+1}, ...d_{l}) iff ∃ã, ã′ = ã′_{1}...ã′_{n}(θ^{∗} ∧ V_{0} ∧ V_{1} ∧ V_{2})

where, θ^{∗} is defined by: θ^{∗}(a, ã, ã′, d_{1}, ...d_{m}, c_{1}, e_{1}, ...c_{n}, e_{n}, d_{m+1}, ...d_{l}) ≡

θ(a, ã, d_{1}, ...d_{l}) ∧ ⋀_{i=1}^{n} ψ_{i}(ã′_{i}, e_{i}) ∧ (ã′_{i} = g_{i}(f(a), d_{1}, ...d_{m}, c_{1}, e_{1}, ...c_{i}))

V_{0} is defined by:

V_{0}(ã, d_{1}, ...d_{m}) iff ⋀_{i=0}^{m}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}...d_{i})

V_{1} is defined by:

V_{1}(ã′, d_{1}, ...d_{m}, c_{1}, e_{1}, ...c_{n}, e_{n}) iff ⋀_{i=1}^{n}(ψ_{i}(ã′_{i}, w_{i}) forks over d_{1}d_{2}...d_{m}c_{1}e_{1}...c_{i}),

and V_{2} is defined by:

V_{2}(ã, d_{1}, ..d_{m}, c_{1}, e_{1}, ..c_{n}, e_{n}, d_{m+1}, ..d_{l}) iff ⋀_{i=m+1}^{l}(ϕ_{i}(ã, y_{i}) forks over d_{1}d_{2}..d_{m}c_{1}e_{1}..c_{n}e_{n}d_{m+1}..d_{i}).

Note that V^{∗} is a pre-τ̃_{st}^{f}-set relation. Thus

U^{∗} = {a| ∃d_{1}, ..d_{m}, c_{1}, e_{1}, ..c_{n}, e_{n}, d_{m+1}, ..d_{l} V^{∗}(a, d_{1}, ..d_{m}, c_{1}, e_{1}, ..c_{n}, e_{n}, d_{m+1}, ..d_{l})}

is a τ̃_{st}^{f}-set. By the construction of a^{∗}, d_{1}^{∗}, ..d_{m}^{∗}, c_{1}^{∗}, e_{1}^{∗}, ..c_{n}^{∗}, e_{n}^{∗}, d_{m+1}^{∗}, ..d_{l}^{∗}, U^{∗} ≠ ∅. By the definition of U^{∗}, U^{∗} ⊆ U ∩(C^{s}\S_{f,n}) (note that if a ∈ U^{∗}, then there are d_{1}, ..., d_{m} ∈ C such that SU_{se}(f(a)/d_{1}...d_{m}) ≥ n and thus by Remark 9.2, SU_{se}(f(a)) ≥ n). So, the proof of Subclaim 9.4 is complete, and thus so is the proof of the theorem.

Recall that a theory T has the wnfcp(=weak non finite cover property) if for each L-formula ϕ(x, y), the D_{ϕ}-rank is finite and definable (the D_{ϕ}-rank of a formula ψ(x, a) is defined by: D_{ϕ}(ψ(x, a)) ≥ 0 if ψ(x, a) is consistent; D_{ϕ}(ψ(x, a)) ≥ α + 1 if for some b, D_{ϕ}(ψ(x, a) ∧ ϕ(x, b)) ≥ α and ϕ(x, b) divides over a; and for limit δ, D_{ϕ}(ψ(x, a)) ≥ δ if it is ≥ α for all α < δ).

Corollary 9.5 Let T be a countable imaginary simple unidimensional theory. Then T is low and thus has the wnfcp.

Proof: By Fact 2.4, T has bounded finite SU-rank in any given sort. Thus the global D-rank in any given sort has a finite bound. Now, let ϕ(x, y) ∈ L. Then ϕ(x, y) is low in x iff Sup{D(x = x, ϕ(x, y), k)| k < ω} < ω. So, clearly every ϕ(x, y) is low in x. Thus T is low. By Corollary 3.8 the extension property is first-order in any unidimensional theory. We conclude T has the wnfcp (see [BPV], Corollary 4.6).

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