Canonizing structural Ramsey theorems

TL;DR

This paper advances structural canonical Ramsey theory by establishing new canonical Ramsey theorems for classes like finite tournaments, posets, and metric spaces, using category theory to simplify complex proofs.

Contribution

It provides several new canonical Ramsey results for various classes and introduces category theory as a tool to handle proof complexity in this field.

Findings

Canonical Ramsey theorem for finite tournaments

Canonical Ramsey theorem for finite posets with linear extensions

Canonical Ramsey theorem for finite linearly ordered metric spaces

Abstract

At the beginning of 1950's Erd\H os and Rado suggested the investigation of the Ramsey-type results where the number of colors is not finite. This marked the birth of the so-called canonizing Ramsey theory. In 1985 Pr\"omel and Voigt made the first step towards the structural canonizing Ramsey theory when they proved the canonical Ramsey property for the class of finite linearly ordered hypergraphs, and the subclasses thereof defined by forbidden substructures. Building on their results in this paper we provide several new structural canonical Ramsey results. We prove the canonical Ramsey theorem for the class of all finite linearly ordered tournaments, the class of all finite posets with linear extensions and the class of all finite linearly ordered metric spaces. We conclude the paper with the canonical version of the celebrated Ne\v{s}et\v{r}il-R\"odl Theorem. In contrast to the "classical" Ramsey-theoretic approach, in this paper we advocate the use of category theory to manage the complexity of otherwise technically overwhelming proofs typical in canonical Ramsey theory.