A Dual Ramsey Theorem for Finite Ordered Oriented Graphs

TL;DR

This paper establishes a dual Ramsey theorem for finite ordered oriented graphs using surjective homomorphisms, expanding the dual Ramsey theory to a new class of structures with categorical formulation.

Contribution

It introduces the first dual Ramsey result for finite ordered oriented graphs based on surjective homomorphisms, utilizing category theory for formalization.

Findings

Proves a dual Ramsey theorem for finite ordered oriented graphs.

Uses surjective homomorphisms instead of embeddings.

Employs category theory to formalize the dual Ramsey property.

Abstract

In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for "direct" Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. 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.