The externally definable Ramsey property and fixed points on type spaces

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for relational structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for all \(n \in \mathbb {N} \), every subflow of the space \(S_n(M)\) of n-types has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.