Related papers: Superstability from categoricity in abstract eleme…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum…
A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence…
We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. $Theorem$: Let $K$ be an AEC with a monster model. - If $K$ has a simple independence relation, then $K$ does not have the…
We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider {\bf Definition.} Let $M_0 \prec…
In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $\mu$ in $\lambda$-categorical abstract elementary classes with no maximal models, where $\lambda$…
We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition know as tameness (currently…
We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not…
We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory…
In this paper, we study a stability transfer theorem in d-tame Metric Abstract Elementary classes, in a similar way as in [BaKuVa], but using superstability-like assumptions which involves a new independence notion (Tame Independence)…
We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric…
We deal with stability theory for ``reasonable'' non-elementary classes without any remanents of compactness (like: above Hanf number or definable by L_{omega_1, omega}).
We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of…
In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note…
We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is $d^p$-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we…
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.…
In this paper we give characterizations of the super-stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned class of first-order theories can be regarded as "not…
We show how to build primes models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: $\mathbf{Theorem.}$ Let $K$ be an almost fully good AEC that is categorical in $\text{LS}…
The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong…
In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an…
Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>LS({\bf K})$ be a regular stability cardinal and let $\chi$ be the…