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

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.