On omega-categorical structures with few finite substructures

TL;DR

This paper investigates the growth rates of orbit-counting sequences in omega-categorical structures, establishing optimal bounds for unstable and homogeneous cases, and linking sub-exponential growth to omega-stability.

Contribution

It provides the first optimal bounds for growth rates in unstable omega-categorical structures and connects sub-exponential growth to omega-stability.

Findings

Optimal exponential growth bound of 2 for unstable structures.

Improved bounds for homogeneous structures in finite relational languages.

Reduction of sub-exponential growth analysis to omega-stable structures.

Abstract

We establish new results on the possible growth rates for the sequence (f_n) counting the number of orbits of a given oligomorphic group on unordered sets of size n. Macpherson showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola's bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences (f_n) of sub-exponential growth reduces to the omega-stable case.