Twenty years of Ne\v{s}et\v{r}il's classification programme of Ramsey classes

TL;DR

This paper reviews twenty years of progress in classifying Ramsey classes, highlighting key developments, open problems, and recent extensions to areas like EPPA and big Ramsey structures within structural Ramsey theory.

Contribution

It provides a comprehensive overview of the classification program of Ramsey classes initiated by Nešetřil, including recent advancements and open problems.

Findings

Identification of numerous Ramsey classes following Nešetřil-R"odl theorem

Connections established between Ramsey classes, Fra"issé classes, and topological dynamics

Recent extensions to EPPA and big Ramsey structures

Abstract

In the 1970s, structural Ramsey theory emerged as a new branch of combinatorics. This development came with the isolation of the concepts of the $\mathbf{A}$-Ramsey property and Ramsey class. Following the influential Ne\v{s}et\v{r}il-R\"odl theorem, several Ramsey classes have been identified. In the 1980s, Ne\v{s}et\v{r}il, inspired by a seminar of Lachlan, discovered a crucial connection between Ramsey classes and Fra\"iss\'e classes, and, in his 1989 paper, connected the classification programme of homogeneous structures to structural Ramsey theory. In 2005, Kechris, Pestov, and Todor\v{c}evi\'c revitalized the field by connecting Ramsey classes to topological dynamics. This breakthrough motivated Ne\v{s}et\v{r}il to propose a program for classifying Ramsey classes. We review the progress made on this program in the past two decades, list open problems, and discuss recent extensions to new areas, namely the extension property for partial automorphisms (EPPA), and big Ramsey structures.