We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. Theorem 0.1Let K be an AEC with a monster model.•If K has a simple independence relation, then K does not have the 2-tree property.•If K has a simple independence relation with the (<ℵ0)-witness property for singletons, then K does not have the tree property. Theorem 0.1 Let K be an AEC with a monster model. If K has a simple independence relation, then K does not have the 2-tree property. If K has a simple independence relation with the (<ℵ0)-witness property for singletons, then K does not have the tree property. The proof of both facts is done by finding cardinal bounds to classes of small Galois-types over a fixed model that are inconsistent for large subsets. We think that this finer way of counting types is an interesting notion in itself. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

Abstract In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent: (1) For all high-enough λ,Khas no long splitting chains. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ. (3) For all high-enough λ,Khas a unique limit model of cardinality λ. (4) For all high-enough λ,Khas a superlimit model of cardinality λ. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Abstract We prove a categoricity transfer theorem for tame abstract elementary classes. Suppose thatKis a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Letλ ≥ Max{χ, LS(K+}.If K is categorical in λ and λ+, then K is categorical in λ++. Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes: Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Letμ0≔ Hanf(K).If and K is categorical in some then K is categorical in μ for allμ .

Abstract Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-transferable or (κ(T), λ)-transferable for all λ. Further if for some μ ≥ |T|,2μ > μ+, stability is equivalent to for all μ ≥ |T|, saturation is (μ, 2μ)-transferable.

Abstract Basic results on the model theory of substructures of a fixed model are presented. The main point is to avoid the use of the compactness theorem, so this work can easily be applied to the model theory of Lω1,ω and its relatives. Among other things we prove the following theorem: Let M be a model, and let λ be a cardinal satisfying λ∣L(M)∣ = λ. If M does not have the ω-order property, then for every A ⊆ M, ∣A∣ ≤ λ, and every I ⊆ M of cardinality λ+ there exists J ⊆ I cardinality λ+ which is an indiscernible set over A. This is an improvement of a result of S. Shelah.

Abstract Let M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is realized in M is realized in N. We discuss the existence of such submodels. The following are corollaries of the existence theorems. (1) If M is of cardinality at least ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1. (2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.

Abstract We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than min(2λ, ℶ2) rich models of cardinality λ (λ > ∣T∣) is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ1 and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.

Abstract Let L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)). Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p. Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular. Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.

In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not. It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC. This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω-saturated models of an ω-stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω-stable theories.

Abstract We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L∞,ω to , ω. The simplest instance is: Theorem 1. Let λ > κ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every X ⊆ M there exists a model N ≺ M containing the set X of power ∣X∣ · κ such that for every pair of finite sequences a, b ∈ N The following theorem is an application: Theorem 2. Let λ<κ, T ∈ , ω, and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, , ω)-unsuperstability property, then T has the (χ, , ω)-unsuperstability property.

Abstract Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now define Our main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω}. This concept is interesting because of Theorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and . If then (∀χ > κ)I(χ, T) = 2χ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). Many years ago the second author proved that . Here we continue that work by proving Theorem 2. . Theorem 3. For everyκ ≤ λwe have . For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. Theorem 4. For every T ⊆ Lκ +, ω, and any set of formulas ⊆ Lκ +, ω such that T ⊇ Lκ +, ω, if T is ( , μ)-unstable for μ satisfyingμμ*(λ,κ) = μ then T is -unstable (i.e. for every χ ≥ λ, T is ( , χ)-unstable). Moreover, T is Lκ +, ω-unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers.