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

Abstract

We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil-Rödl 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.

91 pages, 21 figures. Accepted to Advances in Mathematics. Reformatted to match journal recommendations. Changed numbering of Theorems. Theorem 2.1 in the previous draft is now Theorem 2.11. Theorem 2.2 in the previous draft is now Theorem 2.18

References in corpus (3)

• Ramsey properties and extending partial automorphisms for classes of finite structures
• A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind
• Automorphism groups and Ramsey properties of sparse graphs

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