Superfilters, Ramsey theory, and van der Waerden's Theorem |
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Authors: | Nadav Samet Boaz Tsaban |
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Institution: | aDepartment of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel;bDepartment of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel |
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Abstract: | 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. |
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Keywords: | Superfilters Ramsey theory van der Waerden Theorem Ramsey Theorem Schur Theorem Folkman– Rado– Sanders Theorem Rado Theorem Arithmetic progressions |
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