TY - JOUR

T1 - Superfilters, Ramsey theory, and van der Waerden's Theorem

AU - Samet, Nadav

AU - Tsaban, Boaz

PY - 2009/10/1

Y1 - 2009/10/1

N2 - Superfilters are generalizations of ultrafilters, and 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 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 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.

AB - Superfilters are generalizations of ultrafilters, and 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 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 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.

KW - Arithmetic progressions

KW - Folkman-Rado-Sanders Theorem

KW - Rado Theorem

KW - Ramsey Theorem

KW - Ramsey theory

KW - Schur Theorem

KW - Superfilters

KW - van der Waerden Theorem

UR - http://www.scopus.com/inward/record.url?scp=69249244192&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2009.04.014

DO - 10.1016/j.topol.2009.04.014

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AN - SCOPUS:69249244192

SN - 0166-8641

VL - 156

SP - 2659

EP - 2669

JO - Topology and its Applications

JF - Topology and its Applications

IS - 16

ER -