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

We prove the Ramsey property of 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 author’s Ramsey lift of bowtiefree graphs. We use this framework to give new examples of Ramsey classes. Among others, we show (and characterise) the Ramsey property 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 Structural the Ramsey Theorem. We also show the Ramsey Theorem for structures with linear ordering on relations (“totally ordered structures”). All of these results are natural, easy to state, yet proofs involve most of the theory developed here. We characterise classes of structures defined by forbidden homomorphisms having a Ramsey lift and extend this to special cases of classes with closures. We apply this to prove the Ramsey property of many Cherlin-Shelah-Shi classes. In several instances our results are the best possible and confirm the meta-conjecture that Ramsey classes are “close” to lifted universal classes.