The externally definable Ramsey property and fixed points on type spaces

TL;DR

The paper introduces and develops the externally definable Ramsey property (EDRP) for countable ultrahomogeneous structures and proves a dynamical characterization: for such a structure $M$ with countable age, EDRP is equivalent to the fixed points on type spaces property (FPT). Building on QE and KPT-type ideas, the authors show that FPT implies EDRP and, conversely, that EDRP implies FPT under countable age, extending KPT-type correspondences beyond $oldsymbol{ ext{aleph}_0}$-saturated cases. They develop a robust framework for externally definable colorings, simplify via QE, and apply it to a broad spectrum of Fraïssé structures, including lexicographic products, to demonstrate both the reach and limits of the EDPR–FPT correspondence. The work links model-theoretic definability with topological dynamics, providing new tools to classify Fraïssé structures by their dynamical properties and offering a suite of explicit examples (e.g., rainbow graphs, $S(2),S(3)$, NR-free amalgamations) that illustrate when EDPR holds or fails. The results pave the way for further exploration of definable Ramsey properties and their dynamical signatures in automorphism groups, with potential impact on Preductions of amenability and Ellis group analyses.

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.

Theorems & Definitions (107)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Fraïssé, Fra54
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Definition 2.10
• Definition 2.12
• Lemma 2.13
• Lemma 2.14