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

TL;DR

This paper introduces an abstract framework for finite Ramsey theory, proving a general Ramsey-type theorem that encompasses classical, dual, and new self-dual Ramsey theorems through iterative applications.

Contribution

It presents a novel abstract approach that unifies and generalizes various finite Ramsey theorems, including a new self-dual Ramsey theorem.

Findings

Established a general Ramsey-type theorem applicable to multiple classical results.

Derived the classical Ramsey, Hales--Jewett, and Graham--Rothschild theorems as special cases.

Proved a new self-dual Ramsey theorem extending existing dual results.

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.