Abstract elementary classes stable in $\aleph_0$
Saharon Shelah Sebastien Vasey


  We study abstract elementary classes (AECs) that, in ℵ_0, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ℵ_0. More precisely, there is a superlimit model of cardinality ℵ_0 and the class generated by this superlimit has a type-full good ℵ_0-frame (a local notion of nonforking independence) and a superlimit model of cardinality ℵ_1. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.


Irreducible Convex Paving for Decomposition of Multi-dimensional Martingale Transport Plans

    Hadrien De Marchhadrien.de-march@polytechnique.org. Nizar Touzinizar.touzi@polytechnique.edu.
    Accepted ???. Received ???; in original form December 30, 2023
=================================================================================================





§ INTRODUCTION



 §.§ Motivation

In <cit.> (a revised version of which appears as <cit.>, from which we cite), the first author introduced abstract elementary classes (AECs): a semantic framework generalizing first-order model theory and also encompassing logics such as _ω_1, ω. He studied _ℵ_0-representable AECs (roughly, AECs which are reducts of a class of models of a first-order theory omitting a countable set of types) and generalized and improved some of his earlier results on _ω_1, ω <cit.> and _ω_1, ω (Q) <cit.>.

For example, fix a _ℵ_0-representable AEC  and assume that it is categorical in ℵ_0. Assuming 2^ℵ_0 < 2^ℵ_1 and 1 ≤ (, ℵ_1) < 2^ℵ_1, the first author shows (without even assuming _ℵ_0-representability) <cit.> that  has amalgamation in ℵ_0. Further, <cit.>, it has a lot of structure in ℵ_0 and assuming more set-theoretic assumptions as well as few models in ℵ_2,  has a superlimit model in ℵ_1 <cit.>. This means roughly (see Section <ref>) that there is a saturated model in ℵ_1 and that the union of an increasing chain of type ω consisting of saturated models of cardinality ℵ_1 is saturated.

The reader can think of the existence of a superlimit in ℵ_1 as a step toward showing that the models of cardinality ℵ_1 behave in a “superstable-like” way. Indeed several recent works <cit.> have connected superlimits with other definitions of superstability in AECs, including uniqueness of limit models and local character of orbital splitting.

Another notable consequence of the existence of a superlimit in ℵ_1 is that it implies that there is a model of cardinality ℵ_2. This ties back to a result of the first author: <cit.>: for a _ℵ_0 AEC, categoricity in ℵ_0 and ℵ_1 implies the existence of a model in ℵ_2. The argument first establishes, using only categoricity in ℵ_0 and few models in ℵ_1, that there is a pair M, N of models in _ℵ_1 such that M  N and then uses, in essence, that (by ℵ_1-categoricity) these models are superlimits. In this context, the very strong hypotheses make it possible to avoid referring to any stability-theoretic notions. Still, in more complicated frameworks the existence of a superlimit model in ℵ_1 can be thought of as a key conceptual step toward proving existence of models in higher cardinality and more generally developing a stability theory cardinal by cardinal.

The arguments for the results from <cit.> discussed in the second paragraph of this introduction are complicated by the lack of ℵ_0-stability: one can only get that there are ℵ_1-many orbital types over countable models. The workaround there is to redefine the ordering (but not the class of models) to obtain a stable class, see <cit.>. If the AEC is “nicely-presented”, e.g. a class of models of an _ω_1,ω-sentence or more generally a finitary AEC <cit.>, then this difficulty does not occur (see <cit.>): ℵ_0-stability follows from few models in ℵ_1 and 2^ℵ_0 < 2^ℵ_1. One can also obtain ℵ_0-stability by starting with only countably-many models in ℵ_1 <cit.>. Finally, it is worth noting that (assuming amalgamation and joint embedding in ℵ_0), ℵ_0-stability is upward absolute for _ℵ_0-AECs <cit.>.



 §.§ Main results


The bottom line is that ℵ_0-stability holds in several cases of interest. In fact, there are no known examples which (under 2^ℵ_0 < 2^ℵ_1) are categorical in ℵ_0, have few models in ℵ_1, and are not ℵ_0-stable (see <cit.>). Thus in the present paper, we start with stability in ℵ_0 (and often amalgamation and categoricity in ℵ_0). Our goal is to say as much as we can on the structure of the class, in particular to get superstable-like behavior in ℵ_0 and ℵ_1, without assuming a non-ZFC hypothesis or (, ℵ_1) < 2^ℵ_1.

One of our first results (Theorem <ref>) is that ℵ_0-stability (together with amalgamation and ℵ_0-categoricity) imply that the class  is already _ℵ_0-representable. We also show that the assumption of categoricity in ℵ_0 is not really needed: without assuming it, one can find a superlimit in ℵ_0 and change to the class generated by that superlimit, which will be categorical in ℵ_0. In fact, we prove (Theorem <ref>) that one can characterize brimmed models (also called limit models in the literature) as those that are homogeneous for orbital types. This has as immediate consequence that the brimmed model of cardinality ℵ_0 is superlimit (Corollary <ref>). This last result sheds light on an argument of Lessmann <cit.> and answers a question of Fred Drueck (see footnote 3 on <cit.>), who asked when this equality held. The argument works more generally assuming only density of amalgamation bases, as in <cit.>.

For the main result of this paper, we assume that orbital types over countable models are determined by their finite restrictions. The study of statements of the form “orbital types are determined by their small restrictions” was pioneered by Grossberg and VanDieren <cit.>, who called this condition tameness. Hyttinen and Kesälä <cit.> were the first to specifically study orbital types over finite sets and the condition that they determine orbital types over countable models. 

Following the first author's terminology <cit.>, we call this last condition (<ℵ_0, ℵ_0)-locality (not to be confused with sequence locality <cit.>, which is called locality in <cit.> or <cit.>). This is known to hold for several classes of interest:

   
  
  * Let  be a finitary AEC (see <cit.>; this includes classes of models of _ω_1,ω-sentences) and assume that  is stable in ℵ_0 (finitary AECs have amalgamation and no maximal models by definition). By <cit.>,  is (<ℵ_0, ℵ_0)-local. 
  
  * Finitary AECs are not the only setup where (<ℵ_0, ℵ_0)-locality holds. For example, it is known for quasiminimal pregeometry classes (that may not be finitary <cit.>), see <cit.>, and more generally in the finite U-rank (FUR) classes of Hyttinen and Kangas <cit.> (we thank Will Boney for pointing us to that reference). 
  


We prove the following:


  Let  be an AEC with () = ℵ_0 and countable vocabulary. Assume that  is categorical in ℵ_0,  is (<ℵ_0, ℵ_0)-local,  has amalgamation and no maximal models in ℵ_0 and  is stable in ℵ_0. Then:

  
  
  * (Theorem <ref>) There is a good ℵ_0-frame on _ℵ_0.
  
  * (Corollary <ref>) There is a superlimit model of cardinality ℵ_1.
  


The good ℵ_0-frame (or the superlimit in ℵ_1) imply the nontrivial corollary that  has a model of cardinality ℵ_2 <cit.>. This consequence also follows from a theorem of the second author <cit.> (which however does not give a good ℵ_0-frame or a superlimit in ℵ_1). The conclusion that there is a superlimit model in ℵ_1 seems new, even for finitary AECs or FUR classes.

It is natural to ask whether the locality hypothesis in Theorem <ref> is really needed[In fact, an earlier version of the present paper asserted that it could be derived from the other hypotheses but the argument contained a mistake.]. In fact we do not even know whether the existence of a good ℵ_0-frame implies (<ℵ_0, ℵ_0)-locality:


  Let  be an AEC with () = ℵ_0. If there is a good ℵ_0-frame on , is  (<ℵ_0, ℵ_0)-local?


In Section <ref>, we give a partial answer: any AEC that is ℵ_0-stable, ℵ_0-categorical, and supersimple (in a sense generalizing that of homogeneous model theory <cit.>) is (<ℵ_0,ℵ_0)-local. This generalizes the proof of <cit.>, which shows that quasiminimal pregeometry classes are (<ℵ_0, ℵ_0)-local (see also <cit.>). Supersimple ℵ_0-stable AECs are also much more general than FUR classes.



 §.§ Notes


This paper is organized as follows. Section <ref> gives some background definitions and fixes the notation.  Section <ref> is a technical section on good frames (possibly on uncountable models) which sets up the machinery to prove the main theorem (more precisely, to prove a strong symmetry property for nonforking in good frames). Section <ref> works with countable models and shows that ℵ_0-stability implies the existence of a superlimit in ℵ_0. Section <ref> builds the good ℵ_0-frame and proves the main theorem. Finally, Section <ref> studies a sufficient condition to get (<ℵ_0, ℵ_0)-locality.

This paper was started while the second author was working on a Ph.D. thesis under the direction of Rami Grossberg at Carnegie Mellon University and he would like to thank Professor Grossberg for his guidance and assistance in his research in general and in this work specifically. We also thank Will Boney, Marcos Mazari Armida, and the referee, for their valuable comments on earlier versions of this paper.

Note that at the beginning of several sections, we make global hypotheses assumed throughout the section.



§ PRELIMINARIES


We assume familiarity with the basics of AECs, as presented for example in <cit.>, or the first three sections of Chapter I together with the first section of Chapter II in <cit.>. We also assume familiarity with good frames (see <cit.> or <cit.>; it would help the reader to have a copy of both available during the reading of Section <ref>). This section mostly fixes the notation that we will use.

Given a τ-structure M, we write |M| for its universe and M for its cardinality. We may abuse notation and write e.g a ∈ M instead of a ∈ |M|. We may even write ∈ M instead of ∈<ω|M|.

We write = (K, ) for an AEC. We may abuse notation and write M ∈ instead of M ∈ K. For a cardinal λ, we write _λ for the AEC restricted to its models of size λ. As shown in <cit.>, any AEC is uniquely determined by its restriction _≤ ().

When we say that M ∈ is an amalgamation base, we mean (as in <cit.>) that it is an amalgamation base in _M, i.e. we do not require that larger models can be amalgamated.

For M_0 ∈, we say that M is universal over M_0 if M_0  M and for any N ∈ with M_0  N, if N≤M_0 +  (), there exists f: N  M (usually we will require also that M = M_0). We say that M is (λ, δ)-brimmed over M_0 (often also called (λ, δ)-limit e.g. in <cit.>) if δ < λ^+ is a limit ordinal, M_0 ∈_λ, and there exists an increasing continuous chain N_i : i ≤δ of members of _λ such that N_0 is universal over M_0, N_δ = M, and N_i + 1 is universal over N_i for all i < δ. We say that M is brimmed over M_0 if it is (M, δ)-brimmed over M_0 for some limit δ < M^+. We say that M is brimmed if it is brimmed over some M_0.

The following key concept appears in <cit.>:


  We say that M ∈ is superlimit if, letting λ := M, we have that λ≥ (), M is universal in _λ (i.e. any M' ∈_λ embeds into M), M is not maximal, and whenever δ < λ^+ is limit, M_i : i < δ is increasing with M_i ≅ M for all i < δ, then ⋃_i < δ M_i ≅ M.


The following notion of types already appears in <cit.>. It is called Galois types by many, but we prefer the term orbital types here. They are the same types that are defined in <cit.>, but we also define them over sets. As pointed out in <cit.>, this causes no additional difficulties. The following technical point is important: when the AEC does not have amalgamation, we may want to compute orbital types only in the subclass of amalgamation bases in  (as in <cit.>). Thus we allow orbital types to be computed in a subclass of  in the definition.


  Fix an AEC  and a subclass ^∗ of , closed under isomorphisms.

  
  
  * We say (A, N_1, _1) ^^∗ (A, N_2, _2) if:
    
    
  * For ℓ = 1,2, N_ℓ∈^∗, A ⊆ |N_ℓ|, and _ℓ∈<∞|N_ℓ|.
    
  * There exists N ∈^∗ and f_ℓ : N_ℓ N, ℓ = 1,2, such that f_1 (_1) = _2.
    
  
  * ^^∗ is a reflexive and symmetric relation.  Let E^^∗ be its transitive closure.
  
  * Let _^∗ ( / A; N) be the E^^∗-equivalence class of (, A, N).
  
  *  Define ^∗ (A, N), ^∗ (M), ^∗^<ω (M), ^∗^<ω (∅), etc. for the Stone spaces of orbital types, computed inside ^∗. This is defined as in <cit.>. For example, ^∗^<ω (∅) = {_^∗ ( / ∅; N) | N ∈^∗, ∈<ωN}. 
  
  * When ^∗ =, we omit it.
  


Let us say that an AEC  is stable in λ if for any M ∈_λ, | (M)| ≤λ. This makes sense in any AEC, and is quite well-behaved assuming amalgamation and no maximal models (since then it is known that one can build universal extensions). We will often work in the following axiomatic setup, a slight weakening where full amalgamation is not assumed. This comes from the context derived in <cit.>:


  Let  be an AEC and let λ be a cardinal. We say that  is nicely stable in λ (or nicely λ-stable) if:

  
  
  * () ≤λ.
  
  * _λ≠∅.
  
  *  has joint embedding in λ.
  
  * Density of amalgamation bases: For any M ∈_λ, there exists N ∈_λ such that M  N and N is an amalgamation base (in _λ).
  
  * Existence of universal extensions: For any amalgamation base M ∈_λ, there exists an amalgamation base N ∈_λ such that M  N and N is universal over M.
  
  * Any brimmed model in _λ is an amalgamation base.
  

  We say that  is very nicely stable in λ if in addition it has amalgamation in λ. 


  An AEC  is very nicely stable in λ if and only if () ≤λ, _λ≠∅,  is stable in λ, and _λ has amalgamation, joint embedding, and no maximal models. In particular, stability is a consequence of the existence of universal extensions in Definition <ref>.


We will repeatedly use the following fact <cit.>. 


  Let  be nicely stable in λ and let M_0, M_1, M_2 ∈_λ. Let δ_1, δ_2 < λ^+ be limit ordinals such that δ_1 = δ_2.

  
  
  * If M_ℓ is (λ, δ_ℓ)-brimmed over M_0, for ℓ = 1,2, then M_1 ≅_M_0 M_2.
  
  * If M_ℓ is (λ, δ_ℓ)-brimmed, for ℓ = 1,2, then M_1 ≅ M_2.
  


  The first is a straightforward back and forth argument and the second follows from the first using joint embedding.


  Uniqueness of brimmed models when δ_1≠δ_2 is a much harder property to establish, akin to superstability. See for example <cit.>. However when λ = ℵ_0 we automatically have that δ_1 = δ_2 = ω. 


Good frames were first defined by the first author in his paper number 600, which eventually appeared as Chapter II of <cit.>. The idea is to provide a localized (i.e. only for base models of a given size λ) axiomatization of a forking-like notion for a “nice enough” set of 1-types.  These axioms are similar to the properties of first-order forking in a superstable theory. Jarden and the first author (in <cit.>) later gave a slightly more general definition, not assuming the existence of a superlimit model and dropping some of the redundant clauses. We will make use of good frames for types of finite length (not just length one). Their definition is just like for types of length one, we call them good (<ω, λ)-frames. For the convenience of the reader, we give the full definition from <cit.> here:


  Let λ be an infinite cardinal. A good (<ω, λ)-frame is a triple (, , ) satisfying, where:

  
  
  *  is an abstract elementary class with λ≥ (), _λ≠∅.
  
  * ⊆⋃_M ∈_λ^<ω (M). Moreover, if (a_i : i < n / M; N) ∈ (M), then a_i ∉ M for all i < n.
  
  *  is a relation on quadruples of the form (M_0, M_1, , N), where M_0  M_1  N, ∈<ωN, and M_0, M_1, N are all in K_λ. We write (M_0, M_1, , N) or M_0M_1N instead of (M_0, M_1, a, N) ∈. 
  
  * The following properties hold:
    
      
  * Invariance: If f: N ≅ N' and M_0M_1N, then f[M_0]f()f[M_1]N'. If ( / M_1; N) ∈ (M_1), then (f () / f[M_1]; N') ∈ (f[M_1]).
      
  * Monotonicity: If M_0M_1N, ' is a subsequence of , M_0  M_0'  M_1'  M_1  N'  N  N” with ' ∈ N', and N”∈ K_, then M_0''M_1'N' and M_0''M_1'N”. If ( / M_1; N) ∈ (M_1) and ' is a subsequence of , then (' / M_1; N) ∈ (M_1). [This property and the previous one show that  is really a relation on types. Thus if p ∈^<ω (M_1) is a type, we say p does not fork over M_0 if M_0M_1N for some (equivalently any)  and N such that p =  ( / M_1; N). Note that this depends on , but  will always be clear from context.]
      
  * Nonforking types are basic: If MMN, then ( / M; N) ∈ (M).
        
      
  * K_λ has amalgamation, joint embedding, and no maximal models.
      
  * bs-Stability: | (M)| ≤M for all M ∈ K_λ.
      
  * Density of basic types: If M  N are in K_λ, then there is a ∈ N such that (a / M; N) ∈ (M).
  
  * Existence of nonforking extension: If m ≤ n < ω, p ∈ (M) ∩^m (M), N  M is in K_λ, then there is some q ∈(N) ∩^n (M) that does not fork over M and extends p.
  
  * Uniqueness: If p, q ∈^<ω (N) do not fork over M and p ↾ M = q ↾ M, then p = q.
  
  * Symmetry: If M_0_1M_2N, _2 ∈<αM_2, and (_2 / M_0; N) ∈ (M_0), then there is M_1 containing _1 and N'  N such that M_0_2M_1N'.
  
  * Local character: If δ is a regular cardinal, M_i ∈ K_λ : i ≤δ is increasing continuous, and p ∈ (M_δ) is such that ℓ (p) < δ, then there exists i < δ such that p does not fork over M_i.
  
  * Continuity: If δ is a limit ordinal, M_i ∈ K_λ: i ≤δ and α_i < α : i ≤δ are increasing and continuous, and p_i ∈ (M_i) for i < δ are such that j < i < δ implies p_j = p_i  M_j, then there is some p ∈ (M_δ) such that for all i < δ, p_i = p  M_i.  Moreover, if each p_i does not fork over M_0, then neither does p.
  
  * Transitivity: If M_0  M_1  M_2, p ∈ (M_2) does not fork over M_1 and p ↾ M_1 does not fork over M_0, then p does not fork over M_0.
    

    A good λ-frame is defined similarly, except we require all types to be types of singletons (i.e. they are in (M) instead of ^<ω (M)). We say that an AEC  has a good (<ω, λ)-frame if there is a good (<ω, )-frame where K is the underlying AEC.
  
  
    If = (, , ) is a good-(<ω, λ)-frame (or a good λ-frame), write _ :=. Also write ^ for  and _ = _λ. We will also write M  N as a shortcut for M  N and M, N ∈_ (= _λ).
  


  The reader might wonder about the reasons for having a special class of basic types. Following <cit.>, let us call a good frame type-full if the basic types are all the nonalgebraic types. There are no known examples of a good λ-frame which which cannot be extended to a type-full one. However a construction of good frames of the first author <cit.> builds a non type-full good frame and it is not clear that it can be extended to a type-full one until a lot more machinery has been developed. Thus it can be easier to build a good frame than to build a type-full one, and most results about frames already hold in the non-type-full context. That being said, readers would not miss the essence of the present paper if they assumed that all the frames here were type-full.


  Any good λ-frame (i.e. for types of length one) extends to a good (<ω, λ)-frame (using independent sequences, see <cit.>) or <cit.>. This frame will however not be type-full.


From now on until the end of Section <ref>, “nonforking” will refer to nonforking in a fixed frame  (usually clear from context).



§ WEAK NONFORKING AMALGAMATION


In this section, we work in a good λ-frame and study a natural weak version of nonforking amalgamation, _ ( stands for “left weak nonforking amalgamation”). The goal is to obtain a natural criteria for proving the existence of a superlimit in ℵ_1 and also prepare the ground for the proof of symmetry in the good frame built in Section <ref>. The main results are the existence property (Theorem <ref>) and how the symmetry property of _ is connected to  being  (Theorem <ref>). Throughout this section, we assume:

   
  
  * = (, , ) is a fixed good (<ω, λ)-frame, except that it may not satisfy the symmetry axiom.
  
  *  is categorical in λ.
  


  In this section, λ is allowed to be uncountable. However the case λ = ℵ_0 is the one that will interest us in the next sections.


The reason for not assuming symmetry is that we will use some of the results of this section to prove that the symmetry axiom holds of a certain nonforking relation in Section <ref>.

We will use:

[II.4.3 in <cit.>]
  Let δ < λ^+ be a limit ordinal divisible by λ. Let M_i : i ≤δ be increasing continuous in _. If for any i < δ and any p ∈ (M_i), there exists λ-many j ∈ [i, δ) such that the nonforking extension of p to M_j is realized in M_j + 1, then M_δ is brimmed over M_0.


To understand the definition below, it may be helpful to think of  as type-full. Then _ (M_0, M_1, M_2, M_3) holds if and only if the type of any finite subsequences of M_1 over M_2 does not fork over M_0 (M_3 is the ambient model). Thus _ is an attempt to extend nonforking to types of infinite sequences so that it keeps a strong finite character property. In the present paper, _ will be a helpful technical tool but it is not clear that it has the uniqueness property (in contrast with the relation  from <cit.> or <cit.>, which will have the uniqueness property but requires more assumptions on the good frame). If _ does have the uniqueness property, this has strong consequence on the structure of the frame, see Theorem <ref>.


  Define the following 4-ary relations on _:

  
  
  * _ (M_0, M_1, M_2, M_3) if and only if M_0  M_ℓ M_3 for ℓ = 1,2 and for any ∈<ω|M_1|, if (, M_2, M_3) and (, M_0, M_3) are basic then (, M_2, M_3) does not fork over M_0.
  
  * _ (M_0, M_1, M_2, M_3) if and only if _ (M_0, M_2, M_1, M_3) [ stands for “right weak nonforking amalgamation”].
  
  * _ (M_0, M_1, M_2, M_3) if and only if both _ (M_0, M_1, M_2, M_3) and _ (M_0, M_1, M_2, M_3) [ stands for “weak nonforking amalgamation].
  

  When  is clear from context, we write , , and .



The following result often comes in handy.


  Let δ < λ^+ be a limit ordinal. Let M_i : i ≤δ, N_i : i ≤δ be increasing continuous in _. Assume that for each i ≤ j < δ, we have that (M_i, N_i, M_j, N_j). If for each i < δ, N_i realizes all the basic types over M_i, then N_δ realizes all the basic types over M_δ.


  Let p ∈ (M_δ). By local character, there exists i < δ such that p does not fork over M_i. In particular, p  M_i is basic. Since N_i realizes all the basic types over M_i, there exists a ∈ |N_i| such that p  M_i =  (a, M_i, N_i). Because for all j ∈ [i, δ), (M_i, N_i, M_j, N_j), we have by continuity that (a, M_δ, N_δ) does not fork over M_i, hence by uniqueness it must be equal to p. Therefore a realizes p, as needed.


Next, we investigate the properties of . We are especially interested in the symmetry property: whether  is equal to . To understand it better, we consider the following ordering, defined similarly to ≤_λ^+^∗ from <cit.>:


  For R ∈{, , }, define a relation R on _λ^+ as follows. For M^0, M^1 ∈_λ^+, M^0 R M^1 if and only if there exists increasing continuous resolutions M_i^ℓ∈_λ : i < λ^+ of M^ℓ for ℓ = 0,1 such that for all i < j < λ^+, R (M_i^0, M_i^1, M_j^0, M_j^1).


The following is a straightforward “catching your tail argument”, see the proof of <cit.> (this assumes that all types are basic, but the argument goes through without this restriction). Roughly, it says that if M  N ( is the usual order on ), then we can find a resolution of M and N so that the pieces are in left weak nonforking amalgamation.


  Let M, N ∈_λ^+. If M  N, then M  N.


Whether M  N can be concluded as well seems to be a  much more complicated question, and in fact is equivalent to  being  (Theorem <ref>), a weakening of symmetry. We now observe that an increasing union of a -increasing chain of saturated models is saturated:


  Let δ < λ^++ be a limit ordinal. If M_i : i < δ is a -increasing sequence of saturated models in _λ^+, then ⋃_i < δ M_i is saturated.


  If δ≥λ^+, then any ⋃_i < δ M_i will be λ^+-saturated on general grounds. Thus assume without loss of generality that δ = δ < λ^+. Let M_δ := ⋃_i < δ M_i. We build M_i, j : i ≤δ, j ≤λ^+ such that:

  
  
  * For any i ≤δ, M_i, λ^+ = M_i.
  
  * For any i < δ, j < λ^+, M_i, j∈_.
  
  * For any i ≤δ, M_i, j : j < λ^+ is increasing and continuous.
  
  * For any j ≤λ^+, M_i, j : i < δ is increasing and M_δ, j = ⋃_i < δ M_i, j.
  
  * For any i_1 < i_2 ≤δ, j_1 < j_2 ≤λ^+, M_i_2, j_2 realizes all the types in (M_i_1, j_1).
  

  This is easy to do. Now for each i_1 < i_2 < δ, we have by assumption that M_i_1 M_i_2. Thus the set C_i_1, i_2 of j < λ^+ such that for all j' ∈ [j, λ^+), (M_i_1, j, M_i_2, j, M_i_1, j', M_i_2, j') is a club (that it is closed follows from the local character and continuity axioms of good frames). Therefore C := ⋂_i_1 < i_2 < δ C_i_1, i_2 is also a club. Hence by renaming without loss of generality for all i_1 < i_2 < δ and all j ≤ j' < λ^+, (M_i_1, j, M_i_2, j, M_i_1, j', M_i_2, j').

    Now let N  M_δ be such that N ∈_λ. We want to see that any type over N is realized in M_δ. By Fact <ref>, it is enough to show that any basic type over N is realized in M_δ.

    Let j < λ^+ be big-enough such that N  M_δ, j. It is enough to see that any basic type over M_δ, j is realized in M_δ, j + 1. To see this, use Lemma <ref> with M_i : i ≤δ, N_i : i ≤δ there standing for M_i, j : i ≤δ, M_i, j + 1 : i ≤δ here. We know that for each i ≤ i' < δ, (M_i, j, M_i', j, M_i, j + 1, M_i', j + 1) and therefore (M_i, j, M_i, j  +1, M_i', j, M_i', j + 1). Thus the hypotheses of Lemma <ref> are satisfied.


The next fact will be used to prove the existence property of . Its proof is a direct limit argument similar to e.g. <cit.>. Roughly, the nonforking relation there is given by “there exists a smaller submodel over which the type does not split”; in fact, these smaller submodels have to be kept as part of the data of the tower. This is not needed here. The argument is also similar to <cit.>. However there the symmetry axiom axiom is needed: there is an extra requirement on the type of a certain element b, but here we do not make that requirement so do not need symmetry.


  Let α < λ^+. Let M_i : i ≤α be -increasing continuous (in _λ) and let _i : i < α be given such that _i ∈<ωM_i + 1 for all i < α and (_i, M_i, M_i + 1) ∈ (M_i) (we allow the _i's to have different length).

  There exists N_i : i ≤α -increasing continuous such that:

  
  
  * M_i  N_i for all i ≤α.
  
  * (_i, N_i, N_i + 1) does not fork over M_i.
  


We can now list and then prove some basic properties of weak nonforking amalgamation. For the convenience of the reader, we repeat Hypothesis <ref>.


  Let = (, , ) be a fixed good (<ω, λ)-frame, except that it may not satisfy the symmetry axiom. Assume that  is categorical in λ. Let R ∈{, , }.
  
  
  
  * Invariance: If R (M_0, M_1, M_2, M_3) and f: M_3 ≅ M_3', then R (f[M_0], f[M_1], f[M_2], M_3').
  
  * Monotonicity: If R (M_0, M_1, M_2, M_3) and M_0  M_ℓ'  M_ℓ for ℓ = 1,2, then R (M_0, M_1', M_2', M_3).
  
  * Ambient monotonicity: If R (M_0, M_1, M_2, M_3) and M_3  M_3', then R (M_0, M_1, M_2, M_3'). If M_3” M_3 contains |M_1| ∪ |M_2|, then R (M_0, M_1, M_2, M_3”).
  
  * Continuity: If δ < λ^+ is a limit ordinal and M_i^ℓ : i ≤δ are increasing continuous for ℓ < 4 with R (M_i^0, M_i^1, M_i^2, M_i^3) for each i < δ, then R (M_δ^0, M_δ^1, M_δ^2, M_δ^3).
  
  * Long transitivity: If α < λ^+ is an ordinal, M_i : i ≤α, N_i : i ≤α are increasing continuous and (M_i, N_i, M_i + 1, N_i + 1) for all i < α, then (M_0, N_0, M_α, N_α).
  
  * Existence: If R ≠, M_0  M_ℓ, ℓ = 1,2, then there exists M_3 ∈_λ and f_ℓ: M_ℓ M_3 such that R (M_0, f_1[M_1], f_2[M_2], M_3).
  


  Invariance and the monotonicity properties are straightforward to prove. Continuity and long transitivity follow directly from the local character, continuity, and transitivity properties of good frames. We prove existence via the following claim:

  Claim: There exists N_0, N_1, N_2, N_3 ∈_ such that (N_0, N_1, N_2, N_3) and N_ℓ is brimmed over N_0 for ℓ = 1,2.

  Existence easily follows from the claim: given M_0  M_ℓ, ℓ = 1,2, there is (by categoricity in λ) an isomorphism f: M_0 ≅ N_0 and (by universality of brimmed models) embeddings f_ℓ : M_ℓ→ N_ℓ extending f for ℓ = 1,2. After some renaming, we obtain the desired -amalgam. To obtain an -amalgam, reverse the role of M_1 and M_2.

  Proof of Claim: The idea of the proof is as follows: for some suitable ordinal α, we want to build M_i : i ≤α, a_i ∈ M_i + 1 : i < α with the following property: whenever N_i : i ≤α is as described by Fact <ref> (plus slightly more), we must have that (M_0, M_α, N_0, N_δ), M_α is brimmed over M_0, and N_0 is brimmed over M_0. To achieve this, we simply start with an arbitrary M_i : i ≤α, a_i : i < α and, if it fails the property, take a witness to the failure, add some more a_j's to make it more brimmed, and start again to consider whether this witness satisfies the property. After doing this for sufficiently many steps, we eventually succeed to build the desired object. This is somewhat similar to the construction of a reduced tower in <cit.>, although here we are dealing with nonforking independence and not just set-theoretic disjointness.

  We now start with the proof. Let δ := λ·λ. We choose (M̅^α,a̅^α) by induction on α≤δ such that:
  
  
  
  * ^α = M^α_i :i ≤α is -increasing continuous.

  
  * ^α = _i : i < α, and _i ∈ M_i + 1^α for all i < α.
    
  * For all i < α, (^α_i, M^α_i, M^α_i+1) ∈ (M_i^α).
    
  * For each i ≤δ, M^α_i:α∈ [i,δ] is -increasing continuous.
    
  *  For each i < δ and each α∈ (i, δ], (_i, M_i^α, M_i + 1^α) does not fork over M_i^i.

  *  If p ∈ (M_i^α) for i ≤α < δ, then for λ-many β∈ [α,δ), ( _β,M^β + 1_β,M^β +1_β + 1) is a nonforking extension of p.

  *  If i < α < δ and (, M_0^α, M_i + 1^α) ∈ (M_0^α), then for some β∈ (α,δ) exactly one of the following occurs:
  
  
  *  (a̅,M^β +1_0,M^β+1_i+1) forks over M^α_0.
  
  *  There is no M^*_j:j ≤ i+1 -increasing continuous such that:
    
    
  * M_j^β M_j^* for all j ≤ i + 1.
    
  * (_j, M_j^*, M_j + 1^*) does not fork over M_j^β for all j < i + 1.
    
  * (, M_0^*, M_i + 1^*) forks over M^β_0.
    
  


  
  
   This is possible:
    Along the construction, we also build an enumeration (_j^γ, k_j^γ, i_j^γ, α_j^γ) : j < λ, γ < λ such that for any γ∈ (0, λ), any α < λ·γ, any i < α, any k ≤ i, and any ∈<ωM_i + 1^α, if (, M_k^α, M_i + 1^α) ∈ (M_k^α), then there exists j < λ so that _j^γ =, i_j^γ = i, k_j^γ = k, and α_j^γ = α. We require that always k_j^γ≤ i_j^γ < α_j^γ < λ·γ and the triple (_j^γ, M_k_j^γ^α_j^γ, M_i_j^γ + 1^α_j^γ) represents a basic type. We make sure that at stage λ· (γ + 1) of the construction below, _j^γ', k_j^γ', i_j^γ', α_j^γ' are defined for all j < λ, γ' ≤γ.
    
    For α = 0, take any M_0^0 ∈_. For α limit, let M_i^α := ⋃_β∈ [i, α) M_i^β for i < α and M_α^α := ⋃_i < α M_i^α. Now assume that ^α, ^α have been defined  for α < δ. We define ^α + 1 and _α. Fix ρ and j < λ such that α = λ·ρ + j. We consider two cases.

      
      
  * Case 1: ρ is zero or a limit: Use Fact <ref> to get M_i^α + 1 : i ≤α -increasing continuous such that M_i^α M_i^α + 1 for all i ≤α, and for all i < α, (_i, M_i^α + 1, M_i + 1^α + 1) does not fork over M_i^α. Pick any M_α + 1^α + 1 with M_α^α + 1 M_α + 1^α + 1 and any _α∈<ωM_α + 1^α + 1 such that (_α, M_α^α + 1, M_α + 1^α + 1) ∈ (M_α^α + 1).
      
  * Case 2: ρ is a successor: Say ρ = γ + 1. Let := _j^γ, α_0 := α_j^γ, k_0 := k_j^γ, i_0 := i_j^γ. There are two subcases.
        
        
  * Subcase 1: Either k_0 ≠ 0, or k_0 = 0 and (<ref>) holds with i, α, β there standing for i_0, α_0, α here.

          In this case, we proceed as in Case 1 to define M_i^α + 1 : i ≤α. Then we pick _α, M_α + 1^α + 1 such that (_α, M_α^α, M_α + 1^α + 1) is the nonforking extension of (, M_i_0^α_0, M_i_0 + 1^α_0).

        
  * Subcase 2: k_0 = 0 and (<ref>) fails.

          In this case, let M_j^∗ : j ≤ i_0 + 1 witness the failure and set M_j^α + 1 := M_j^∗ for j ≤ i_0 + 1. Then continue as in Case 1 and define _α, M_α + 1^α + 1 as before.
        
      
    
   This is enough:

      We proceed via a series of subclaims:

      Subclaim 1: If p ∈ (M_i^δ) for i < δ, then for λ-many β∈ [i,δ), ( _β,M_β^δ,M^δ_β + 1) is a nonforking extension of p.

        Proof of subclaim 1: Pick i' ∈ (i, δ) such that p does not fork over M_i^i'. By (<ref>), we know that for λ-many β∈ [i', δ), the nonforking extension of p  M_i^i' to M_β^β + 1 is realized in M_β + 1^β + 1 by _β. But by (<ref>) we also have that (_β, M_β^δ, M_β + 1^δ) does not fork over M_β^β. In particular by uniqueness _β also realizes p. †_subclaim 1

          Subclaim 2: M_δ^δ is brimmed over M_0^δ.
          
          Proof of subclaim 2: Apply Fact <ref> to the chain M_i^δ : i ≤δ, using the previous step. †_subclaim 2

          We now choose ^* = M_i^∗ : i ≤δ increasing continuous such that M_0^∗ is brimmed over M_0^δ, M_i^δ M_i^∗ for all i ≤δ, and (_i, M_i^∗, M_i + 1^∗) does not fork over M_i^δ. This is possible, see case 1 above. Now let (N_0, N_1, N_2, N_3) := (M_0^δ, M_δ^δ, M_0^∗, M_δ^∗). We have just said that M_0^∗ is brimmed over M_0^δ, and by subclaim 2, M_δ^δ is brimmed over M_0^δ. Thus N_ℓ is brimmed over N_0 for ℓ = 1,2. It remains to see:

          Subclaim 3: (M_0^δ, M_δ^δ, M_0^∗, M_δ^∗)

          Proof of subclaim 3: Pick ∈<ωM_δ^δ such that (, M_0^δ, M_δ^δ) is basic. By local character, there exists α < δ such that (, M_0^δ, M_δ^δ) does not fork over M_0^α. Further, we can increase α if necessary and pick i < α such that ∈<ωM_i + 1^α. We now apply Clause (<ref>). We know that (<ref>) fails for all β∈ (α, δ) by the choice of α, therefore (<ref>) must hold for all β∈ (α, δ). Now if (, M_0^∗, M_δ^∗) forks over M_0^δ, then it must fork over M_0^β for all high-enough β, but then M_j^∗ : j ≤ i + 1 would contradict Clause (<ref>). Therefore (, M_0^∗, M_δ^∗) does not fork over M_0^δ, as desired. †_subclaim 3

          †_Claim
  


The following properties of  may or may not hold in general (we have no examples for the failure of symmetry, but uniqueness fails in the last good frame of the Hart-Shelah example, see <cit.>):


  Let R ∈{, , }.

  
  
  * We say that R has the symmetry property if R (M_0, M_1, M_2, M_3) implies R (M_0, M_2, M_1, M_3).
  
  * We say that R has the uniqueness property if whenever R (M_0, M_1, M_2, M_3) and R (M_0, M_1, M_2, M_3'), there exists M_3” with M_3'  M_3” and f: M_3  M_3”.
  


The following are trivial observations about the definitions:

   
  
  *  has the symmetry property, and  has the symmetry property if and only if  has the symmetry property if and only if =  =.
  
  *  has the uniqueness property if and only  has it.
  


Recall from <cit.>:


   is  when the following is impossible:

  There exists an increasing continuous M_i : i < λ^+, N_i : i < λ^+, a basic type p ∈ (M_0), and _i : i < λ^+ such that for any i < λ^+:

  
  
  * M_i  N_i.
  
  * _i + 1∈ |M_i + 2| and (_i + 1, M_i + 1, M_i + 2) is a nonforking extension of p, but (_i + 1, N_0, N_i + 2) forks over M_0.
  
  * ⋃_j < λ^+ M_j is saturated.
  


We now show that being  is a consequence of symmetry for . Moreover,  allows us to build a superlimit in λ^+.


  (<ref>) ⇒ (<ref>) ⇔ (<ref>) ⇒ (<ref>), where:

  
  
  *   has the symmetry property.
  
  *   is .
  
  *  For M, N ∈_λ^+ both saturated, M  N implies M  N.

  
  *  There is a superlimit model in _λ^+.
  

   
    
  * (<ref>) implies (<ref>): This follows from Lemma <ref> and the fact that the saturated model in λ^+ is universal and has a proper extension <cit.>.

    
  * (<ref>) implies (<ref>): Fix a witness M_i : i < λ^+, N_i : i < λ^+, _i : i < λ^+, p to the failure of being . Write M_λ^+ := ⋃_i < λ^+ M_i, N_λ^+ := ⋃_i < λ^+ N_i. By assumption, M_λ^+ is saturated. Clearly, increasing the N_i's will not change that we have a witness so without loss of generality N_λ^+ is also saturated. We claim that M_λ^+ N_λ^+. We show this by proving that for any i < λ^+ and any j ≤ i + 1, (M_j, N_j, M_i + 2, N_i + 2). Indeed, (_i + 1, N_j, N_i + 2) forks over M_j: if not, then by transitivity (_i + 1, N_j, N_i + 2) does not fork over M_0, and hence (_i + 1, N_0, N_i + 2) does not fork over M_0, and we know that this is not the case of the witness we selected.

    
  * (<ref>) implies (<ref>): Fix M, N saturated in _λ^+ such that M  N but M  N. 

      Claim: For any A ⊆ |M| of size λ, there exists M_0  M_1  M and N_0  N_1  N such that M_0  N_0, M_1  N_1, A ⊆ |M_0|, but (M_0, N_0, M_1, N_1).

      Proof of Claim: If not, we can use failure of the claim and continuity of  to build increasing continuous resolution M_i : i ≤λ^+, N_i : i ≤λ^+ of M and N respectively such that (M_i, N_i, M_j, N_j) for all i < j < λ^+. Thus M  N, contradicting the assumption. †_Claim

      Build M_i^∗ : i ≤λ^+, N_i^∗ : i ≤λ^+ increasing continuous resolutions of M, N respectively such that for all i < λ^+, M_i^∗ N_i^∗ and (M_i + 1^∗, N_i + 1^∗, M_i + 2^∗, N_i + 2^∗). This is possible by the claim. Let _i + 1^∗∈ |M_i + 2^∗| witness the -forking, i.e. (_i + 1^∗, N_i + 1^∗, N_i + 2^∗) forks over M_i + 1^∗. By Fodor's lemma, local character, and stability, there exists a stationary set S, i_0 < λ^+ and p ∈ (M_i_0^∗) such that for all i ∈ S, (_i + 1^∗, M_i^∗, M_i + 2^∗) is the nonforking extension of p. Without loss of generality, i_0 is limit and all elements of S are also limit ordinals.

      Now build an increasing continuous sequence of ordinals j_i : i < λ^+ as follows. Let j_0 := i_0. For i limit, let j_i := sup_k < i j_k. For i successor, pick any j_i ∈ S with j_i > j_i - 1.

      Now for i not the successor of a limit, let M_i := M_j_i^∗, N_i := N_j_i^∗, _i := _j_i^∗. For i = k + 1 with k a limit, set M_i := M_j_k^∗, N_i := N_j_k^∗, _i := _j_i^∗. This gives a witness to the failure of being .
    
  * (<ref>) implies (<ref>): If  has the symmetry property, then by Remark <ref>, =  =. By Fact <ref>, it follows that M  N implies M  N for any M, N ∈_λ^+, so (<ref>) holds.
  



  Are the conditions in Theorem <ref> all equivalent?


  Is there a good λ-frame  such that _ does not have the symmetry property?


The next result shows that the uniqueness property has strong consequences. The first author has given conditions under which when λ = ℵ_0, failure of uniqueness implies nonstructure <cit.>.


  Assume that  is a good (<ω, λ)-frame (so it satisfies symmetry). If  has the uniqueness property, then  has the symmetry property and  is successful  (see <cit.>).


  By <cit.> (used with the pre-(≤λ, λ)-frame induced by , recalling Fact <ref>)  is weakly successful. This implies that there is a relation = _ that is a nonforking relation respecting  (see <cit.>, in particular it has all the properties listed in Theorem <ref>, as well as uniqueness and symmetry). Now as  respects , we must have that (M_0, M_1, M_2, M_3) implies (M_0, M_1, M_2, M_3). Since  has the uniqueness property and  has the existence property, it follows from <cit.> that =. In particular,  has the symmetry property.

  To see that  is successful , it is enough to show that for M, N ∈_λ^+, M  N implies M  N (where  is defined as in Definition <ref>). This is immediate from Fact <ref> and =.


To prepare for the proof of symmetry in the λ = ℵ_0 case, we end this section by introducing yet another notion of nonforking amalgamation ( stands for “very weak nonforking amalgamation”). In this case, we look at finite sequences both on the left and the right hand side. We show that if  is a good frame, then  has the symmetry property and locality of types implies that =. Thus in this case  has the symmetry property too.

   
  
  * For M  N, B ⊆ |N|, ∈<ωN, we say that (, B, N) does not fork over M if there exist M', N' with N  N', M  M'  N', and B ⊆ |M'| such that (, M', N') does not fork over M_0.
  
  * We define a 4-ary relation _ = on _ by (M_0, M_1, M_2, M_3) if and only if M_0  M_ℓ M_3, ℓ = 1,2 and for any ∈<ωM_1 and any finite B ⊆ |M_2|, if (, M_0, M_3) and (, M_2, M_3) are both basic, then (, B, M_3) does not fork over M_0.
  


   Assume that  is a type-full good (<ω, λ)-frame.
  
  
  *  has the symmetry property: (M_0, M_1, M_2, M_3) if and only if (M_0, M_2, M_1, M_3).
  
  * If for any M ∈_ and any p ≠ q ∈ (M) there exists B ⊆ |M| finite such that p  B ≠ q  B, then =. In particular,  has the symmetry property.
  

   
  
  * By the symmetry axiom of good frames.
  
  * This is observed in <cit.>. In details, it suffices to show that for M  N, p ∈ (N) does not fork over M if and only if p  B does not fork over M for all finite B ⊆ |N|. Let q ∈ (N) be the nonforking extension of p  M. For any finite B ⊆ |N|, we have that q  B = p  B, by the uniqueness property for (the extended notion of) forking, see <cit.>. Therefore by the assumption we must have p = q, as desired.
  




§ BUILDING A SUPERLIMIT


In this section, we work in ℵ_0 and show assuming ℵ_0-stability and amalgamation that  is _ℵ_0 (Theorem <ref>) and has a superlimit (Corollary <ref>).

 
= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary).


We will use without comments Fact <ref> and Remark <ref>. The essence of it is that since λ = ℵ_0 all brimmed models have the same length, and hence are isomorphic (and the isomorphism fixes any common base they may have).

First note that if  is stable and has few models, we can say something about its definability:


  Assume that (, ℵ_0) ≤ℵ_0.
  
  
  
  * The set {M ∈ K_ℵ_0: |M| ⊆ω} is Borel.
  
  * If  has amalgamation in ℵ_0 and is stable in ℵ_0, then the set {(M,N):M  N and |N| ⊆ω} is Σ^1_1.
  

  In particular if  has amalgamation in ℵ_0 and is stable in ℵ_0, then  is a _ℵ_0-representable AEC.

   
  
  * Fix M ∈_ℵ_0. By Scott's isomorphism theorem, there exists a formula ϕ_M of _ℵ_1, ℵ_0 (τ_) such that N ϕ_M if and only if M ≅ N. Now observe that the set
    
    
    {N : N  is a τ_-structure with |N| ⊆ω and  N ϕ_M}


    is Borel and use that (, ℵ_0) ≤ℵ_0.
  
  * For M, N ∈_ℵ_0 with M  N, let us say that N is almost brimmed over M if either N is brimmed over M, or N is -maximal. Using amalgamation, it is easy to check that if N, N' are both almost brimmed over M, then N ≅_M N' (as in Fact <ref>, recalling that the chains witnessing the brimmedness must have cofinality ω). Moreover there always exists an almost brimmed model over any M ∈_ℵ_0.

    Fix M_n^∗ : n < ω such that for any M ∈_ℵ_0 there exists n < ω such that M ≅ M_n^∗ (possible as (,ℵ_0) ≤ℵ_0). For each n < ω, fix N_n^∗∈_ℵ_0 almost brimmed over M_n^∗. We have:

    
      
  ⊛_1 For M, N ∈_ℵ_0:

        
        
  * There is n < ω and an isomorphism f: M_n^∗≅ M.
        
  * If N is almost brimmed over M, then any such f extends to g: N_n^∗≅ N.
        
    

    
    
  ⊛_2
      For M_1, M_2 ∈_ℵ_0, M_1  M_2 if and only if M_1 ⊆ M_2 and for some n_1, n_2 < ω, for some (N,f_1,f_2) we have: M_1 ⊆ M_2 ⊆ N
      and for ℓ = 1,2, f_ℓ is an isomorphism from (M_n_ℓ^*,N_n_ℓ^*) onto (M_ℓ,N).
    

[Why?  The implication “if" holds by the coherence axiom of AECs.  The implication
“only if" holds as there is N ∈_ℵ_0 which is almost brimmed over
M_2 (and so M_2  N) hence N is almost brimmed over M_1 
  and use ⊛_1 above.]


The result now follows from ⊛_2.
  

  By <cit.>, it follows that  is _ℵ_0.


We now study homogeneous models and show that they coincide with brimmed models. Note that the homogeneity here is with respect to a set D of orbital types.


  Let D be a set of orbital types over the empty set and let M ∈. We say that M is (D, ℵ_0)-homogeneous if it realizes all the types in D and whenever p ∈ D is the type of an (n + m)-elements sequence and ∈n M realizes p^n (the restriction of p to its first n “variables”), there exists a sequence ∈mM such that  realizes p.


The next result characterizes the countable brimmed model in AECs that are nicely stable in ℵ_0 (recall Definition <ref>).


  Assume that  is nicely stable in ℵ_0 and let ^∗ be the class of amalgamation bases in _ℵ_0. Let M ∈_ℵ_0. The following are equivalent:

  
  
  * M is brimmed.
  
  * M is (^∗^<ω (∅), ℵ_0)-homogeneous (see Definition <ref>(<ref>)).
  



  Let M ∈_ℵ_0. First we show:

  Claim: If M is brimmed, , ∈ M, then _^∗ ( / ∅; M) = _^∗ ( / ∅; M) if and only if there is an automorphism f of M sending  to . 

  Proof of Claim: The right to left direction is clear. Now assume that _^∗ ( / ∅; M) = _^∗ ( / ∅; M). Say M_i : i < ω witness that M is brimmed, and without loss of generality assume that , ∈ M_0. Then _^∗ ( / ∅; M_0) = _^∗ ( / ∅; M_0). Since ^∗ has amalgamation, there exists M_0' ∈^∗ with M_0  M_0' and f: M_0 → M_0' so that f () =. Since M_1 is universal over M_0, we can assume without loss of generality that M_0' = M_1. Now extend f to an automorphism of M using a back and forth argument.

  From the claim, it follows directly that if M is brimmed, then it is (^∗^<ω (∅), ℵ_0)-homogeneous. Conversely, the countable (^∗^<ω (∅), ℵ_0)-homogeneous model is unique (standard back and forth argument) and so it must also be brimmed.


  By adding constants to the language, we can also characterize brimmed models over M_0 as those that are homogeneous for orbital types of finite sequences over M_0.



  If  is nicely stable in ℵ_0, then there is a superlimit model of cardinality ℵ_0.


  Let M ∈_ℵ_0 be brimmed (it exists by nice stability in ℵ_0). We claim that M is superlimit. To see this, we check the conditions of Definition <ref>. On general grounds, brimmed models are universal in _ℵ_0, are not maximal (from the definition of nice stability), and there is a unique brimmed model of cardinality ℵ_0. Still, it is not obvious that if M_i : i < δ is an increasing chain of brimmed models in _ℵ_0 and δ < ω_1, then ⋃_i < δ M_i is brimmed. To see this, we use Theorem <ref>: each M_i is ((^∗^<ω (∅), ℵ_0)-homogeneous, and it is clear from the definition that an increasing union of such homogeneous models is homogeneous. Thus M_δ is (^∗^<ω (∅), ℵ_0)-homogeneous. By Theorem <ref> again, M_δ is brimmed, as desired.
 

We have justified assuming amalgamation in the following sense:


  If  is nicely stable in ℵ_0, then there exists an AEC ' = (K', ') such that:

  
  
  *  (') = ℵ_0.
  
  * '_<ℵ_0 = ∅.
  
  * τ_' = τ_.
  
  * K' ⊆ K and for M, N ∈ K', M ' N if and only if M  N.
  
  *  For any M ∈_ℵ_0 there exists M' ∈_ℵ_0' with M  M'.
  
  *  ' is categorical in ℵ_0.
  
  *  ' is very nicely stable in ℵ_0. In particular it has amalgamation in ℵ_0.
  
  *  ' is _ℵ_0.
  


  Let M ∈_ℵ_0 be superlimit (exists by Corollary <ref>). Let K_ℵ_0' := {N ∈ K : N ≅ M}. Now let ' be the AEC generated by (K_ℵ_0', ) (in the sense of <cit.>). One can easily check that ' is nicely stable in ℵ_0; from this and ℵ_0-categoricity we get amalgamation in ℵ_0, hence (<ref>) holds. As for (<ref>), it follows from Theorem <ref>.




§ BUILDING A GOOD ℵ_0-FRAME


The aim of this section is to build a good ℵ_0-frame from nice ℵ_0-stability. By Corollary <ref>, we may restrict the class to a superlimit so that it is categorical in ℵ_0. As before, we assume:

 
= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary).


The nonforking relation of the frame will be nonsplitting:


  For M ∈_ℵ_0 and A ⊆ |M|, p ∈^<ω (M) splits over A if whenever p =  ( / M; N), there exists _1, _2 ∈ M such that (_1 / A; M) =  (_1 / A; M) but (_1 / A; N) ≠ (_2 / A; N).


The following is proven in <cit.>. Similar proofs appear in <cit.> or <cit.>.


  Assume that  is nicely stable in ℵ_0 and categorical in ℵ_0. If M ∈_ℵ_0 and p ∈^<ω (M), then there exists A ⊆ |M| finite such that p does not split over A.


The following result about nonsplitting will also come in handy. It appears in various forms in the literature, see e.g. <cit.>.


  Assume that  is nicely stable in ℵ_0 and categorical in ℵ_0. Let M  N both be in _ℵ_0, p,q ∈^<ω (N). If both p and q do not split over a finite subset of M and p  A = q  A for all finite A ⊆ |M|, then p  B = q  B for all finite B ⊆ |N|.


  Let N' be brimmed over N. Let _1, _2 ∈ N' realize p and q respectively. Fix A ⊆ |M| finite such that p and q do not split over A. Let B ⊆ |N| be finite and let  be an enumeration of B. Since M is brimmed, there exists ' ∈ M such that ( / A; N) =  (' / A; N). By nonsplitting, (_ℓ/ A; N') =  (_ℓ' / A; N') for ℓ = 1,2. Now since ' ∈ M we have by assumption that p  A ' = q  A '. Therefore (_1 ' / A; N') =  (_2 ' / A; N'). Putting these equalities together, (_1  / A; N') =  (_2  / A; N'), so p  B = q  B, as desired.



  Let  be nicely stable in ℵ_0 and categorical in ℵ_0. We define a pre-(<ω, λ)-frame = (^, ,  ^) by:

  
  
  * ^ =.
  
  * For M_0  M  N all in _ℵ_0, n < ω, = a_i :i < n∈nN, M_0MN holds if and only if a_i ∉ M for all i < n and there exists a finite A ⊆ |M_0| so that ( / M; N) does not split over A.
  
  * For M ∈_ℵ_0, ^ (M) is the set of all types of finite sequences a_i : i < n over M such that for all i < n, a_i ∉ M.

  


In order to prove that  is a good ℵ_0-frame, we will make an additional locality hypothesis. See Example <ref> and the next section for setups where it holds.


   is (<ℵ_0, ℵ_0)-local if for any M ∈  p, q ∈^<ω (M), p  A = q  A for all finite A ⊆ |M| implies p = q. We say that  is weakly (<ℵ_0, ℵ_0)-local if this holds for a superlimit M.


  The definition of locality includes types of any finite length, not just of length one. This will be used to prove the symmetry property of _, via Theorem <ref>.


We now prove, assuming nice stability, categoricity, and locality, that the pre-frame defined above is a good ℵ_0-frame.


  Assume that  is nicely stable in ℵ_0 and categorical in ℵ_0. If  is (<ℵ_0, ℵ_0)-local, then  (Definition <ref>) is a type-full good (<ω, ℵ_0)-frame. Moreover _ has the symmetry property (recall Definitions <ref> and <ref>). In particular,  is .


  Once we have shown that  is a type-full good frame, the moreover part follows from Theorem <ref>. The last sentence is by Theorem <ref>.

  Now except for symmetry, the axioms of good frames are easy to check (see the proof of <cit.>). For example:

  
  
  * Local character: Let M_i : i ≤δ be increasing continuous in _. Let p ∈^ (M_δ). By Fact <ref>, there exists a finite A ⊆ |M_δ| such that p does not split over A. Pick i < δ such that A ⊆ |M_i|. Then p does not fork over M_i.
  
  * Uniqueness: by Lemma <ref> and locality.
  
  * Extension: follows on general grounds, see <cit.>.
  
  
  Symmetry is the hardest to prove, and is done as in <cit.>. We give a full proof for the convenience of the reader.

  Suppose that (, N_2, N_3) does not fork over N_0 and let ∈<ωN_2 \ N_1. We want to find N_1, N_3' such that N_0  N_1  N_3', N_3  N_3', ∈<ωN_1 and (, N_1, N_3') does not fork over N_0. Assume for a contradiction that there is no such N_1. Using existence for _ (see Theorem <ref>), as well as the extension property for nonforking, we can increase N_2 and N_3 if necessary and find N_1 such that _ (N_0, N_1, N_2, N_3), N_ℓ is brimmed over N_0, and N_3 is brimmed over N_ℓ for ℓ = 1,2. By assumption, p :=  (, N_1, N_3) forks over N_0.

  Claim 1: Let I be the linear order [0, ∞) ∩ℚ. There exists an increasing chain M_s : s ∈ I such that for any s < t in I, M_s, M_t are in _ℵ_0 and M_t is brimmed over M_s.

    Proof of Claim 1: Let ϕ∈_ω_1, ω be a Scott sentence for the model in _ℵ_0. Let ψ∈_ω_1, ω be a Scott sentence for a pair M, N ∈_ℵ_0 such that N is brimmed over M. Now let K^∗ be the class of sequences M_j : j ∈ J such that J is a linear order, M_j ϕ for all j ∈ J, and (M_j, M_k) ψ for all j < k in J. It is easy to see that K^∗ is axiomatizable by a sentence in _ω_1, ω. Moreover, for each α < ω_1, there is a sequence M_i : i < α in K^∗. By <cit.>, this implies that K^∗ contains an I-indexed member, as desired. †_Claim 1

  Fix I, M_s : s ∈ I as in Claim 1. Fix N_0' such that N_0 is brimmed over N_0' and p  N_0 does not fork over N_0'.

  For any fixed infinite J ⊆ I, write M_J := ⋃_s ∈ J M_s. Assume now that M_I is brimmed over M_J. Let N_0^J := M_J, N_1^J := M_I. Let N_3^J be brimmed over N_1^J. By categoricity and uniqueness of brimmed models, there exists f_0 : N_0' ≅ M_0, f_0^J : N_0 ≅ N_0^J, f_1^J : N_1 ≅ N_1^J, and f_3^J : N_3 ≅ N_3^J such that f_0 ⊆ f_0^J ⊆ f_1^J ⊆ f_3^J. Let f_2^J := f_3^J  N_2 and let N_2^J := f_2^J[N_2]. Note that _ (N_0^J, N_1^J, N_2^J, N_3^J) holds.

  Let p_J :=  (f_3^J (), f_3^J[N_1], f_3^J[N_3]) =  (f_3^J (), M_I, N_3^J). Since we are assuming that (, N_1, N_3) forks over N_0, we have that p_J forks over N_0^J. Moreover p_J  N_0^J does not fork over M_0.

  Claim 2: If J has no last elements, I \ J has no first elements, and t ∈ I \ J, then p_J  M_t forks over N_0^J.

  Proof of Claim 2: Suppose that p_J  M_t does not fork over N_0^J. Note that M_t is brimmed over M_J. Find N_1' such that N_0  N_1'  N_1, N_1' is brimmed over N_1, and f_1^J : N_1' ≅ M_t. Let ' ∈<ωN_1' be such that (', N_0, N_1') =  (, N_0, N_1). Since _ (N_0, N_1, N_2, N_3), we know that (', N_2, N_3) does not fork over N_0, hence by uniqueness (, N_2, N_3) =  (', N_2, N_3). But we have assumed that (, N_1', N_3) does not fork over N_0 and ' ∈<ω_1N_1', hence by a simple renaming we obtain a contradiction to our hypothesis that symmetry failed. †_Claim 2
  
  Claim 3: If J_1 ⊊ J_2 are both proper initial segments of I with no last elements and J_2 \ J_1 has no first elements, then p_J_1≠ p_J_2.

  Proof of Claim 3: Fix t ∈ J_2 \ J_1. By Claim 2, p_J_1 M_t forks over N_0^J_1. We claim that p_J_2 M_t does not fork over N_0^J_1. Indeed recall that N_0^J_2 = M_J_2 and by assumption p_J_2 N_0^J_2 does not fork over M_0. Therefore by monotonicity also p_J_2 M_t does not fork over M_J_1 = N_0^J_1. †_Claim 3

  To finish, observe that there are 2^ℵ_0 cuts of I as in Claim 3. Therefore stability fails, a contradiction.   


The next corollary does not assume categoricity, but uses amalgamation in ℵ_0, rather than just density of amalgamation bases.


  If  is very nicely stable in ℵ_0 and weakly (<ℵ_0, ℵ_0)-local, then  has a superlimit of cardinality ℵ_1.


  By Corollary <ref>,  has a superlimit N_0 in ℵ_0. Let ' be the class generated by this superlimit, as described by the proof of Corollary <ref>. Then ' is categorical in ℵ_0 and nicely stable in ℵ_0, hence we can apply Theorem <ref> and get a type-full  ℵ_0-frame with underlying class _ℵ_0'. By Theorem <ref>, ' has a superlimit model in ℵ_1. This is also a superlimit in : the only nontrivial property to check is universality. Let M ∈_ℵ_1. Fix any M_0 ∈_ℵ_0 with M_0  M. By universality of N_0, there exists f: M_0 → N_0. Now let N ∈' be superlimit in ℵ_1 with N_0  N. Using amalgamation (amalgamation in ℵ_0 suffices for this, see <cit.>), we can find g: M → N extending f, as needed.






§ LOCALITY FROM SUPERSIMPLICITY


In this section, we give a sufficient condition for locality. As before, we assume:

 
= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary).


In the context of a nicely ℵ_0-stable AEC, the following definition generalizes that of a supersimple homogeneous model <cit.>. The idea is that we want to have a nice notion of nonforking available for all finite sets (not only models). However we do not require that forking over finite sets satisfies any uniqueness requirement. Thus it is not a-priori clear that supersimplicity implies the existence of a good frame (although this will follow from ℵ_0-categoricity and Theorems <ref>, <ref>).

Throughout this section expressions such as “nonforking” or “not fork” will refer to the relation defined in Definition (<ref>)(<ref>) below. We give examples after the definition.


  Assume that  is nicely ℵ_0-stable and categorical in ℵ_0. We say that  is supersimple if there exists a 4-ary relation  such that:

  
  
  * (A, B, C, N) implies that N ∈_ℵ_0, A ∪ B ∪ C ⊆ |N| and A, B, C are all finite or countable. We write ABCN instead of (A, B, C, N). Below, we may abuse notation and write e.g. A_1A_2 instead of A_1 ∪ A_2, or N instead of ABCN, where A, B, C stand for the ranges of , , and  respectively.
  
  * Normality: ABCN if and only if AABACN.
  
  *  Invariance under -embeddings: If f: N → N' and A ∪ B ∪ C ⊆ |N|, then ABCN, if and only if f[A]f[B]f[C]N'. This shows that  is really a relation on types, so we say ( / C; N) does not fork over A if ACN.
  
  * Monotonicity: If ABCN and A ⊆ A' ⊆ B' ⊆ B then A'B'CN. 
  
  * Symmetry: If ABCN, then ACBN.
  
  * Local character: If M ∈_ℵ_0 and p ∈^<ω (M), then there exists A ⊆ |M| finite such that p does not fork over A.
  
  * Extension: If p ∈^<ω (C; N) does not fork over A ⊆ C, then there is q ∈^<ω (N) such that q extends p and q does not fork over A.
  
  * Transitivity: If ABCN and CBDN with A ⊆ C ⊆ D, then ABDN.
  
  * Relationship with splitting: If M  N are both in _ℵ_0, p ∈^<ω (N), and p does not fork over M, then there is A ⊆ |M| finite such that p does not split over A.
  


  It may be helpful to compare Definition <ref> with Definition <ref>. The idea of <ref> is to give a sort of analog of good frames but to allow types over sets. Note that the statement of symmetry in <ref> is more technical, precisely because types over sets are not allowed. However the idea is the same. Another difference is that local character in <ref> is stated as “every type does not fork over a finite set”. In <ref>, it is stated as “every type over the union of an increasing chain does not fork over a previous element of the chain”. Again, the lack of types over sets makes it impossible to state the former in good frames.

   
  
  * Working inside a supersimple homogeneous model N (in the sense of <cit.>), we can define ABCN to hold if and only if B is ℵ_0-free from C over A (in the sense of <cit.>). The first four conditions of Definition <ref> are then easy to check. Symmetry is <cit.> and transitivity is <cit.>. Local character and extension are given by the definition in <cit.>. Now if in addition N is ℵ_0-stable (in the sense of <cit.> or equivalently in the sense given here), then by <cit.> the last axiom of Definition <ref> holds.
  
  * Let  be a FUR class <cit.> (this includes in particular all quasiminimal pregeometry classes). Then letting  be defined as in <cit.>, we can also check that it satisfies Definition <ref>. 
  

    

We first show that if a type p over a finite set does not fork over a subset A of a countable model M, then the type is realized inside M.


  Assume that  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M  N both be in _ℵ_0, and let B ⊆ |N| be finite. Let p ∈^<ω (B; N). If p does not fork over B ∩ |M|, then p is realized in M.


  Extending N if necessary, we can assume without loss of generality that N is brimmed over M. Let q ∈^<ω (N) be a nonforking extension of p. Let  be an enumeration of B. Fix A' ⊆ |M| finite and big-enough such that B ∩ |M| ⊆ A', ( / M; N) does not split and does not fork over A' and q does not split over A' (recall Fact <ref>). Let M_0  M be such that M is brimmed over M_0 and M_0 contains A'. Let ∈ M be such that  realizes q  M_0. We have that M_0MN so by symmetry and monotonicity, M_0N. By extension, we can pick M_0' such that M_0  M_0'  N, M_0' contains , and M_0M_0'N. Now as q  M_0' does not split over a finite subset of M_0 and  realizes q  M_0, we must have by Lemma <ref> that  realizes q  B = p, as desired.


We will prove locality in supersimple ℵ_0-stable AECs by a back and forth argument. More precisely, we start with M  N, N brimmed over M, and elements _1, _2 ∈ N whose types over every finite subset of M match. First, we will do a back and forth argument to find an automorphism of N sending _1 to _2 and fixing M setwise (Lemma <ref>). We will then use this automorphism and nonsplitting to build another automorphism that fixes M pointwise (Theorem <ref>).

The next lemma starts setting up the stage by making sure that we can map an element of M to an element of M.


  Assume that  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M  N both be in _ℵ_0. Let _1, _2 ∈ N, _1, _2, _1 ∈ M be such that (_1 _1; N) =  (_2 _2; N). If _1_1_1N, then there exists _2 ∈ M such that (_1 _1 _1; N) =  (_2 _2 _2; N).


  Extending N if necessary, we can assume without loss of generality that N is brimmed over M. By symmetry, _1_1_1N. Let f be an automorphism of N sending _1 _1 to _2 _2. By invariance, _2f(_1)_2N (but we do not know that f(_1) ∈ M). Let q :=  (f (_1) / _2 _2; N). We have to show that q is realized in M. Since q does not fork over _2 ∈ M, this is exactly what Lemma <ref> tells us.


Our main lemma in the back and forth argument will be:


  Assume that  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M  N both be in _ℵ_0 with N brimmed over M. Let _1, _2 ∈ N, _1, _2 ∈ M be such that (_1 _1; N) =  (_2 _2; N) and _ℓ_ℓMN for ℓ = 1,2. If _1 ∈ N, _1' ∈ M, there exists _1, _2, _2' ∈ M and _2 ∈ N such that:
 
  
  
  * (_1 _1 _1 _1' _1; N) =  (_2 _2 _2 _2' _2; N).
  
  * _ℓ_ℓ_ℓ'_ℓ_ℓMN for ℓ = 1,2.
  


  Using Lemma <ref>, we can enlarge _1 and _2 if necessary to assume without loss of generality that _1' is empty. Now using local character, fix _1 ∈ M such that _1 _1_1 _1MN. By Lemma <ref>, there exists _2 ∈ M such that (_1 _1 _1; N) =  (_2 _2 _2; N). Let f be an automorphism of N witnessing this. By extension and monotonicity, pick _2 ∈ N such that (_2 / _2 _2 _2; N) =  (f (_1) / _2 _2 _2; N) and _2 _2 _2_2M _2N. It remains to see that _2 _2_2 _2MN. We do this using a standard nonforking calculus argument: by normality, _2 _2 _2_2 _2M _2N and we also know from the hypotheses of the lemma that _2_2MN, so by monotonicity and normality _2 _2_2 _2 _2MN. Now using transitivity, monotonicity, and symmetry, _2_2_2 _2MN, as desired.


We can now build the desired automorphism which fixes M setwise.


  Assume that  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M ∈_ℵ_0, A ⊆ |M| finite, p, q ∈^<ω (M) such that p and q do not fork over A. If p  A = q  A, then there exists an automorphism f of M fixing A such that f (p) = q.


  Let N ∈_ℵ_0 be brimmed over M. Say p =  (_1 / M; N), q =  (_2 / M; N). Let  be an enumeration of A and let _ℓ :=, ℓ = 1,2. Now apply Lemma <ref> repeatedly in a back and forth argument to build an automorphism g of N fixing A such that g (_1) = _2 and g[M] = M. Let f := g  M.


We now show that we can actually build an automorphism fixing M pointwise. The next lemma is the main argument for this:


  Assume that  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M ∈_ℵ_0 and let N be brimmed over M. Let _1, _2 ∈ N be such that (_1 / A; N) =  (_2 / A; N) for all finite A ⊆ |M|. For any _1 ∈ N, there exists _2 ∈ N such that (_1 _1 / A; N) =  (_2 _2 / A; N) for all finite A ⊆ |M|.


  Let p :=  (_1 / M; N), q :=  (_2 / M; N). Fix A ⊆ |M| finite such that both p and q do not fork over A and (_1 _1 /M ; N) does not split over A. By Lemma <ref>, there exists an automorphism f of M fixing A such that f (p) = q. Let g be an automorphism of N extending f such that g (_1) = _2 and let _2 := g (_1). We claim that this works. Fix C ⊆ |M| finite and let _2 be an enumeration of C. Let _1 := g^-1 (_2). We know that (_1 / A; N) =  (_2 / A; N), so by nonsplitting, (_1 _1 _1 / A; N) =  (_1 _1 _2 / A; N). Applying g, we have that (_1 _1 _1 / A; N) =  (_2 _2 _2 / A; N). Putting the two equalities together, (_1 _1 _2 / A; N) =  (_2 _2 _2 / A; N), so (_1 _1 / C; N) =  (_2 _2 / C; N), as desired.


We have arrived to the main theorem of this section:


  If  is nicely ℵ_0-stable, supersimple, and categorical in ℵ_0, then  is (<ℵ_0, ℵ_0)-local (recall Definition <ref>).


  Let M ∈_ℵ_0 and let p, q ∈^<ω (M) be such that p  A = q  A for all finite A ⊆ |M|. Let N be brimmed over M and let _1, _2 ∈ N realize p and q respectively. Now apply Lemma <ref> in a back and forth argument to get an automorphism of N fixing M taking _1 to _2.


amsalpha



