2023/07/18 by Barbosa, Keegan Dasilva, Mašulović, Dragan
Computer Science · Mathematics · #05C55 #18A99 #Advanced Topology and Set Theory #Category Theory (math.CT) #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2307.09100
openalex publication_date 2023/07/18 · openalex created_date 2023/07/20 · openalex updated_date 2026/07/28
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.