Finite Structures with Few Types
The monograph develops a structure theory for a class of finite structures, whose description lies on the border between model theory and group theory. Model theoretic methods are applied to a study of certain finite permutation groups, the automorphism groups of structures for a fixed finite language with a bounded number of orbits on 4-tuples. The work uses methods of permutation theory, model theory, classical geometries and combinatorics. The principal results (summarized in Section 1.2) are finiteness theorems showing that the structures under consideration fall naturally into finitely many families, where each family is parameterized by finitely many numerical invariants, dimensions of associated coordinating geometries.