I will introduce two prominent dynamical systems for a given toplogical group, the greatest ambit and the universal minimal flow, as spaces of (near) ultrafilters on certain Boolean algebras. Representing a topological group as a group of isometries of a highly symmetric structure, I will hint how metrizability and triviality of the universal minimal flow is linked to the (approximate) structural Ramsey property. My focus will lie on problems that arise in the study of universal minimal flows in Ramsey theory, model theory, set theory and continuum theory.

There seems to be a natural relationship between topological Ramsey spaces and Fraïssé classes of finite structures. In fact, for some Fraïssé classes satisfying the Ramsey property, it is possible to define a topological Ramsey space such that the Fraïssé limit of the class is essentially an element of the space. We will talk about examples of this phenomenon, describe the general case to some extent, and comment about how this could be understood as a abstract tool to classify Fraïssé structures.