Superfilters, Ramsey theory, and van der Waerden's Theorem

TL;DR

This paper introduces superfilters, a generalization of ultrafilters, to unify and extend classical Ramsey theory results like van der Waerden's Theorem, confirming a conjecture and providing a broad generalization of multiple theorems.

Contribution

It establishes properties of superfilters that unify and extend key Ramsey theoretic theorems, confirming a conjecture and generalizing results to smaller colored sets.

Findings

Confirmed a conjecture of Kočinac and Di Maio.

Generalized classical theorems to smaller colored sets.

Unified multiple Ramsey theorems under superfilters.

Abstract

Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.