Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

Abstract

We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales--Jewett theorem (with Shelah's bounds), the Graham--Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem are all obtained as iterative applications of the general result.