Dual Ramsey theorems for relational structures

TL;DR

This paper establishes explicit dual Ramsey theorems for various finite relational structures using surjective maps and category theory, providing a dual perspective to classical Ramsey results.

Contribution

It introduces a novel approach focusing on morphisms rather than objects to prove dual Ramsey theorems for relational structures.

Findings

Explicit dual Ramsey theorems for finite relational structures

Introduction of surjective maps as morphisms in dual Ramsey context

Extension of Nešetřil-R"odl Theorem to dual setting

Abstract

In this paper we provide explicit dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with an explicit dual of the Ne\v{s}et\v{r}il-R\"odl Theorem for relational structures. Instead of embeddings which are crucial for "direct" Ramsey results, for each class of structures under consideration we propose a special class of surjective maps and prove a dual Ramsey theorem in such a setting. In contrast to on-going Ramsey classification projects where the research is focused on fine-tuning the objects, in this paper we advocate the idea that fine-tuning the morphisms is the key to proving dual Ramsey results. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.