Let C be a model-complete core which is a reduct of a homogeneous Ramsey structure A with finite relational signature. We present characterisations of when the existence of a pseudo-Siggers polymorphism of C implies the existence of a pseudo-Siggers polymorphism of C which is canonical over A. This has applications for the complexity of constraint satisfaction: Barto and Pinsker showed that an ω-categorical model-complete core structure C which does not have a pseudo-Siggers polymorphism has an NP-hard constraint satisfaction problem (CSP). On the other hand, if C is a reduct of a finitely bounded homogeneous structure B and C has a pseudo-Siggers polymorphism which is canonical with respect to B, then the CSP for C can be solved in polynomial time, by a reduction to a finite-domain CSP of Bodirsky and Mottet and the finite-domain dichotomy theorem of Bulatov and Zhuk. Our results allow to re-derive and generalise some of the existing complexity classifications for infinite-domain CSPs, for example for the class of all structures with exponential labelled growth. We also verify the infinite-domain tractability conjecture for first-order expansions of the basic relations of the spatial reasoning formalism RCC5.

Let \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c,d with d<1. We show that \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that \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 \mathcal{K}_{{\operatorname{exp}}{+}}. We also show that Thomas’ conjecture holds for \mathcal{K}_{{\operatorname{exp}}{+}}: all structures in \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

A first-order structure A is called monadically stable iff every expansion of A by unary predicates is stable. In this article we give a classification of the class M of ω-categorical monadically stable structures in terms of their automorphism groups. We prove in turn that M is smallest class of structures which contains the one-element pure set, closed under isomorphisms, and closed under taking finitely disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in M is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in M has finitely many reducts up to interdefinability, thereby confirming Thomas' conjecture for the class M.

Let F ω 2 denote the countably infinite dimensional vector space over the two element field and GL(ω, 2) its automorphism group. Moreover, let Sym(F ω 2 ) denote the symmetric group acting on the elements of F ω 2 . It is shown that there are exactly four closed subgroups, G, such that GL(ω, 2) ≤ G ≤ Sym(F ω 2 ). As F ω 2 is an ω-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of F ω 2 . These reducts are also analyzed. In the last section the closed groups containing the infinite symplectic group Sp(ω, 2) are classified.

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. Our construction over the 2-element field is related to the Reed-Muller codes.