Towards a characterisation of universal categories of relational structures

Answering a conjecture of Konig, Frucht first established that every finite group is isomorphic to the automorphism group of a finite graph. Since then, there has been a series of results regarding the representation of groups in various finite and infinite structures. These culminated in the work of Isbel, who initiated the study of algebraically universal categories, i.e. those that fully embed every category of universal algebras. In this talk, I'll discuss recent work that establishes a partial characterisation of algebraically universal categories of relational structures, given in terms of a sparsity notion known as nowhere density and its model-theoretic consequences. This extends a result of Nesetril-Ossona de Mendez on categories of finite graphs to the context of categories of relational structures of unbounded size.