Tukey reducibility for categories -- In search of the strongest statement in finite Ramsey theory

TL;DR

This paper introduces a categorical framework to compare the strength of finite Ramsey theory statements using a generalized Tukey reducibility via pre-adjunctions, establishing a hierarchy among these statements.

Contribution

It proposes a novel categorical approach to classify finite Ramsey statements based on their Ramsey strength, extending Tukey reducibility to categories.

Findings

Finite Dual Ramsey Theorem is as powerful as Graham-Rothschild Theorem.

Identified the weakest and strongest categories in finite Ramsey theory.

Established a hierarchy of Ramsey statements based on categorical reducibility.

Abstract

Every statement of the Ramsey theory of finite structures corresponds to the fact that a particular category has the Ramsey property. We can, then, compare the strength of Ramsey statements by comparing the ``Ramsey strength'' of the corresponding categories. The main thesis of this paper is that establishing pre-adjunctions between pairs of categories is an appropriate way of comparing their ``Ramsey strength''. What comes as a pleasant surprise is that pre-adjunctions generalize the Tukey reducibility in the same way categories generalize preorders. In this paper we set forth a classification program of statements of finite Ramsey theory based on their relationship with respect to this generalized notion of Tukey reducibility for categories. After identifying the ``weakest'' Ramsey category, we prove that the Finite Dual Ramsey Theorem is as powerful as the full-blown version of the Graham-Rothschild Theorem, and conclude the paper with the hypothesis that the Finite Dual Ramsey Theorem is the ``strongest'' of all finite Ramsey statements.