Counting in Uncountably Categorical Pseudofinite Structures

TL;DR

This paper establishes a polynomial relationship between definable set sizes and minimal subsets in uncountably categorical pseudofinite structures, linking model-theoretic properties to combinatorial class classifications.

Contribution

It demonstrates that definable sets in such structures have polynomial pseudofinite cardinalities related to Morley rank, and classifies certain finite structure classes as polynomial R-mecs and asymptotic classes.

Findings

Definable subsets have polynomial pseudofinite cardinalities.

Finite classes with ultraproducts satisfying the same theory are polynomial R-mecs.

Such classes are also N-dimensional asymptotic classes.

Abstract

We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial $R$-mecs as well as $N$-dimensional asymptotic classes, where $N$ is the Morley rank of the theory.