This paper attempts to give a self-contained development of dividing theory (also called forking theory) in a strongly homogeneous structure. Dividing is a combinatorial property of the invariant relations on a structure that have yielded deep results for the models of so-called "simple" first-order theories. Below we describe for the nonspecialist how this paper fits in the broader context of geometrical stability theory. Naturally, some background in first-order model theory helps to understand these motivating results; however virtually no knowledge of logic is assumed in this paper. Readers desiring a more thorough description of geometrical stability theory are referred to the surveys [Hru97] and [Hru98].Traditionally, geometrical stability theory is a collection of results that apply to definable relations on arbitrary models of a complete first-order theory. It is equivalent and convenient to restrict our attention to the definable relations on a fixed representative model of the theory, called a universal domain. Using the terminology of the abstract, a universal domain is an uncountable model M equipped with the first-order definable relations R which is strongly |M |-homogeneous and compact ; i.e., if { X i : i ∈ I }, where |I| < |M |, is a family of definable relations on M so that i∈F X i = ∅ for any finite F ⊂ I, then i∈I X i = ∅. For our purposes the reader can assume there is a one-to-one correspondence between (complete first-order) theories and universal domains.A massive amount of abstract model theory was developed en route to Shelah's proof of Morley's Conjecture about the number of models, ranging over uncountable cardinals, of a fixed first-order theory [She90]. Most of the work concerned the case of a stable theory, which will not be defined here for the sake of brevity. What is relevant is that most theorems describing the models of a stable theory rely on the forking independence relation. The forking independence relation, F , is a ternary relation on the subsets of the universal domain of a theory, where F (A, B; C) is read "A is forking independent from B over C" (see Remark 2.2). In a stable theory forking independence is symmetric (over C), has finite character (in A and B), bounded dividing, the free extension property and is transitive. (See Definition 2.5, Theorem 2.14 and Corollary 2.15 for precise statements of these properties.) These properties facilitate the introduction of several notions of dimension that lead to procedures for determining when two models are isomorphic. The combinatorial-geometric properties of the definable relations reflected in these dimensions profoundly impact the structure of the models beyond the question of fixing an isomorphism type. The results connected to algebra, known as geometrical

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The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the ℵ0-stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is ℵ0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models).

ABSTRACT. This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:

Abstract. We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.