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Tukey reducibility for categories -- In search of the strongest statement in finite Ramsey theory
Zbornik radova, Tome 22 (2025) no. 30, p. 193

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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.
DOI : 10.64191/zr24040410106
Classification : 05C55, 18A99
Keywords: finite Ramsey theory, Tukey reducibility, categoryinebreak theory 
Keegan Dasilva Barbosa; Dragan Mašulović. Tukey reducibility for categories -- In search of the strongest statement in finite Ramsey theory. Zbornik radova, Tome 22 (2025) no. 30, p. 193 . doi: 10.64191/zr24040410106
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