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Every statement of the Ramsey theory of finite structures corresponds to thefact that a particular category has the Ramsey property. We can, then, comparethe strength of Ramsey statements by comparing the ``Ramsey strength'' of thecorresponding categories. The main thesis of this paper is that establishingpre-adjunctions between pairs of categories is an appropriate way of comparingtheir ``Ramsey strength''. What comes as a pleasant surprise is thatpre-adjunctions generalize the Tukey reducibility in the same way categoriesgeneralize preorders. In this paper we set forth a classification program ofstatements of finite Ramsey theory based on their relationship with respect tothis generalized notion of Tukey reducibility for categories. After identifyingthe ``weakest'' Ramsey category, we prove that the Finite Dual Ramsey Theoremis as powerful as the full-blown version of the Graham-Rothschild Theorem, andconclude the paper with the hypothesis that the Finite Dual Ramsey Theorem isthe ``strongest'' of all finite Ramsey statements.

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