Maximality in first-order theories and the PM_k hierarchy.

In the classification of first-order theories, many "dividing lines" have been defined in order to understand the complexity and the behavior of some classes of theories. These classes are usually defined by forbidding some specific configurations of definable subsets. Patterns (of consistency and inconsistency) are essentially descriptions of abstract configurations of sets and hence can give a general framework to study dividing lines. In this talk, we describe this general framework and we introduce a notion of maximal complexity, first defined by Shelah, by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define Positive Maximality and the PM_k hierarchy, and describe some results about them.