The externally definable Ramsey property and fixed points on type spaces

TL;DR

This paper introduces the externally definable Ramsey property, a weaker form of the classical property, linking it to fixed points in type spaces and exploring its implications in ultrahomogeneous structures.

Contribution

It establishes the equivalence between the externally definable Ramsey property and fixed point properties in type spaces for ultrahomogeneous structures with countable age.

Findings

Externally definable Ramsey property is equivalent to fixed point properties in type spaces.

Basic results analogous to classical Ramsey theory are established.

Examples include analysis of lexicographic products of structures.

Abstract

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for ultrahomogeneous 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 each natural number n, every subflow of the space of n-types with parameters in M has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.