Discrete mathematics seminar - Fraïssé limits of Steiner triple systems

Dates

Wednesday, March 17, 2021 - 14:00 to 15:00

Speaker: Bridget Webb (Open University)

Title: Fraïssé limits of Steiner triple systems

Abstract:

A mathematical structure is homogeneous if every isomorphism between two of its substructures can be extended to an automorphism of the whole. Fraïssé's theorem says that if a countably infinite class of finite structures obeys certain properties, and so is an amalgamation class, then there is a homogeneous countably infinite structure, its Fraïssé limit, whose finitely generated substructures are precisely the elements of the amalgamation class. For example, the Fraïssé limit of the class of all graphs is the well-known Rado graph.

In this talk we will look at homogeneous Steiner triple systems, including some recent work with Daniel Horsley (Monash) where we construct uncountably many homogeneous Steiner triple systems as Fraïssé limits of amalgamation classes of finite Steiner triple systems avoiding specified subsystems. These systems are in some ways analogous to the Hensen graphs, however, unlike the case for graphs, it is unknown whether it is possible to completely classify all homogeneous Steiner triple systems. We will consider future avenues of research that may help shed light on this difficult problem.