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

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 theirRamsey 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 thestrongest'' of all finite Ramsey statements.