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.Comment: Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Green-Tao Theorem on arithmetic progressions in the primes. The bibliography is now update

Ideal version of Ramsey’s theorem

“…He considered in his paper coideals instead of ideals and called them Ramsey coideals. Recently, h-Ramsey ideals have been considered also in [12]. The authors call them Ramsey superfilters and apply them to topological selection principles.…”

Ramsey Theory and the Borel Conjecture

“…The equivalence of ( 2) and ( 7) for n=2 and k=2 is Theorem 8 of [9]. The equivalence of (1) and ( 2) is a result of [44]. The remaining equivalences are then derived as was done above for Ω.…”

Rothberger's property in all finite powers

“…The equivalence of ( 2) and ( 7) for n=2 and k=2 is Theorem 8 of [2]. The equivalence of ( 1) and ( 2) is a result of [11]. The remaining equivalences are then derived as was done above for Ω.…”