Superfilters, Ramsey theory, and van der Waerdenʹs Theorem

Superfilters are generalizations of ultrafilters, and capture the underlying concept in Ramsey-theoretic theorems such as van der Waerdenʹs Theorem. We establish several properties of superfilters, which generalize both Ramseyʹs Theorem and its variants for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kočinac and Di Maio, which is a generalization of a Ramsey-theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindmanʹs 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.