Ramsey Classes: Examples and Constructions

TL;DR

This paper surveys classes of relational structures with the Ramsey property, exploring their fundamental examples, combinatorial proofs, methods for deriving new classes, and open problems in their classification.

Contribution

It provides a comprehensive overview of fundamental Ramsey classes, their properties, and methods to construct new classes, highlighting open problems in their classification.

Findings

Ramsey classes have the amalgamation property and homogeneous Fraisse-limits.

Various methods exist to derive new Ramsey classes from known ones.

Open problems remain in classifying all Ramsey classes.

Abstract

This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial property which resembles the statement of Ramsey's classic theorem. Such classes of structures have been called Ramsey classes. Nesetril and Roedl showed that they have the amalgamation property, and therefore each such class has a homogeneous Fraisse-limit. Ramsey classes have recently attracted attention due to a surprising link with the notion of extreme amenability from topological dynamics. Other applications of Ramsey classes include reduct classification of homogeneous structures. We give a survey of the various fundamental Ramsey classes and their (often tricky) combinatorial proofs, and about various methods to derive new Ramsey classes from known Ramsey classes. Finally, we state open problems related to a potential classification of Ramsey classes.