A footnote to the KPT theorem in structural Ramsey theory

TL;DR

The paper addresses when a Fraïssé class of rigid finite structures has a Fraïssé limit whose automorphism dynamics force a total-order reduct, or else a concrete Ramsey counterexample exists. It offers a direct, constructive proof via ω-categoricity and a 2-element-type orientation mechanism that builds directed graphs encoding inter-type relations, yielding a dichotomy: either a total-order reduct emerges or a Ramsey failure is witnessed by a 2-element A and some B. An explicit example demonstrates that rigid, non-Ramsey Fraïssé classes do exist, clarifying the boundary between rigidity and the Ramsey property in structural Ramsey theory. The results illuminate the interplay between model-theoretic reducts, automorphism groups, and combinatorial partition properties, and open questions about bounds, amenability, and reducts in rigid Fraïssé families.

Abstract

The celebrated theorem of Kechris, Pestov and Todorčević connecting structural Ramsey theory with topological dynamics has as a consequence that the Fraïssé limit of a Ramsey class of non-trivial finite relational structures has a reduct which is a total order; this implies an earlier result of Nešetřil, according to which the structures in such a class are rigid (have trivial automorphism groups). In this paper, we give an alternative proof of this fact. If $\mathcal{C}$ is a Fraïssé class of rigid structures over a finite relational language, then either the Fraïssé limit of $\mathcal{C}$ has a reduct which is a total order, or there is an explicit failure of the Ramsey property involving a pair $(A,B)$ of structures in $\mathcal{C}$ with $|A|=2$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 2.1
• Theorem 3.1
• Proposition 3.2
• Proposition 3.3