All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)

TL;DR

This paper generalizes the structural Ramsey theorem to classes of ordered structures with closures and local properties, solving open problems and providing new examples of Ramsey classes with broad applications.

Contribution

It introduces a unified framework for proving the Ramsey property for complex classes, including those with closures and forbidden homomorphisms, extending previous results.

Findings

Proves Ramsey property for classes with closures and local properties.

Finds Ramsey lifts of convexly ordered S-metric spaces.

Proves Ramsey theorem for finite models with functions and relations.

Abstract

We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Ne\v{s}et\v{r}il-R\"odl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered $S$-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the structural Ramsey theorem. Both of these results are natural, and easy to state, yet their proofs involve most of the theory developed here. We also characterise Ramsey lifts of classes of structures defined by finitely many forbidden homomorphisms and extend this to special cases of classes with closures. This has numerous applications. For example, we find Ramsey lifts of many Cherlin-Shelah-Shi classes.