A footnote to the KPT theorem in structural Ramsey theory

TL;DR

This paper provides an alternative proof that Fraïssé limits of certain classes are either ordered or fail the Ramsey property, deepening understanding of the connection between structural Ramsey theory and topological dynamics.

Contribution

It offers a new proof of a known fact about the structure of Fraïssé limits in the context of Ramsey classes and their automorphism groups.

Findings

Fraïssé limits of rigid structures either have a total order reduct or exhibit explicit Ramsey property failure.

The paper clarifies the relationship between rigidity, ordering, and Ramsey properties in structural classes.

Provides an alternative proof to a key result linking Ramsey theory and topological dynamics.

Abstract

The celebrated theorem of Kechris, Pestov and Todor\v{c}evi\'c connecting structural Ramsey theory with topological dynamics has as a consequence that the Fra\"{\i}ss\'e 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 Ne\v{s}et\v{r}il, 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 Fra\"{\i}ss\'e class of rigid structures over a finite relational language, then either the Fra\"{\i}ss\'e 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 $|A|=2$.