Structures with not too fast unlabelled growth

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