Generalized Amalgamation in Simple Theories and Characterization of Dependence in Non-elementary Classes

We examine the properties of dependence relations in certain non-elementary classes and first-order simple theories. There are two major parts. The goal of the first part is to identify the properties of dependence relations in certain non-elementary classes that, firstly, characterize the model-theoretic properties of those classes; and secondly, allow to uniquely describe an abstract dependence itself in a very concrete way. I investigate totally transcendental atomic models and finite diagrams, stable finite diagrams, and a subclass of simple homogeneous models from this point of view. The second part deals with simple first-order theories. The main topic of this part is investigation of generalized amalgamation properties for simple theories. Namely, we are trying to answer the question of when does a simple theory have the property of n-dimensional amalgamation, where 2-dimensional amalgamation is the Independence theorem for simple theories. We develop the notion of n-simplicity and strong nsimplicity for 1 ≤ n ≤ ω, where both “1-simple” and “strongly 1-simple” is the same as “simple.” We present examples of simple unstable theories in each subclass and prove a characteristic property of n-simplicity in terms of n-dividing, a strengthening of the dependence relation called dividing in simple theories. We prove 3-dimensional amalgamation property for 2-simple theories, and, under an additional assumption, a strong (n + 1)-dimensional amalgamation property for strongly n-simple theories. Stable theories are strongly ω-simple, and the idea behind developing extra simplicity conditions is to show that, for instance, ω-simple theories are almost as nice as stable theories. The third part of the thesis contains an application of ω-simplicity to construct a Morley sequence without the construction of a long independent sequence.