Structures with not too fast unlabelled growth

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 $\mathscr{S}$.