# Ramsey Classes with Closure Operations (Selected Combinatorial Applications)

## TL;DR

This paper establishes the Ramsey property for classes of ordered structures with closures, generalizing many classical results and providing new applications in combinatorics and graph theory.

## Contribution

It introduces a unified framework for Ramsey properties in structures with closures, extending classical theorems and deriving new combinatorial results.

## Key findings

- Proves Ramsey property for ordered sets with equivalences on the power set
- Establishes Ramsey theorems for Steiner systems and resolvable designs
- Provides partial Ramsey results for H-factorizable graphs

## Abstract

We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl Theorem, the author's Ramsey lift of bowtie-free graphs as well as the Ramsey Theorem for Finite Models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of Structural Ramsey Theorem. We give here a more concise reformulation of recent authors paper "All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)" and the main purpose of this paper is to show several applications. Particularly we prove the Ramsey property of ordered sets with equivalences on the power set, Ramsey theorem for Steiner systems, Ramsey theorem for resolvable designs and a partial Ramsey type results for $H$-factorizable graphs. All of these results are natural, easy to state, yet proofs involve most of the theory developed.