Structures with not too fast unlabelled growth

TL;DR

This paper classifies structures with subexponential orbit growth rates, showing they are countably many, interpretable in the rationals, and have finitely many reducts, confirming a conjecture for this class.

Contribution

It provides a complete classification of structures with controlled growth rates in terms of their automorphism groups and confirms Thomas' conjecture for this class.

Findings

Countably many such structures up to bidefinability.

All are first-order interpretable in (Q;<).

Structures have finitely many first-order reducts.

Abstract

Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in $\mathscr{S}$ in terms of their automorphism groups. As a consequence of our classification we show that $\mathscr{S}$ has only countably many structures up to bidefinability, all these structures are first-order interpretable in $(\mathbb{Q};<)$ and they are interdefinable with a finitely bounded homogeneous structure. Furthermore, we also show that all structures in $\mathscr{S}$ have finitely many first-order reduct up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathscr{S}$.