Permutation groups with small orbit growth

Abstract Let 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄{\mathfrak{A}} such that the automorphism group of 𝔄{\mathfrak{A}} has at most c⁢nd⁢n{cn^{dn}} orbits in its componentwise action on the set of n-tuples with pairwise distinct entries, for some constants c,d{c,d} with d<1{d<1}. We show that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}. We also show that Thomas’ conjecture holds for 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}}: all structures in 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} have finitely many first-order reducts up to first-order interdefinability.