Finite Structures with Few Types

I will report on joint work with G. Cherlin on the quasi-finite axiomatizability of smoothly approximable structures, and on finite structures with few types. Let L be a finite language, k an integer, and C(L, k) be the class of finite L-structures with at most k 5-types, The large members of C(L, k) with no nontrivial 0-definable equivalence relation are known to be bi-interpretable with (finite unions of) certain classical geometries. (Work of Cherlin-Lachlan, Kantor-Liebeck-Macpherson.) A dimension of an L-structure M is the dimension of a geometry interpretable in M, such that every automorphism of the geometry lifts to an automorphism of M (with some other conditions.) C(L, k) can be naturally (and effectively) divided into a finite number of families C ϕ (L, k). Within each C ϕ , a finite number of dimensions is identified; every first order statement is equivalent to a Boolean combination of statements asserting that a given dimension is finite (and fixed). In particular, the isomorphism type of a structure in F i is determined by its dimensions. These dimensions can be varied essentially independently. This generalizes Lachlan’s theory of shrinking and stretching homogeneous structures for a finite relational language. The proof involves methods of stability theory (geometries, orthogonality, modularity, stable groups) applied in this unstable context.

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