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

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

2008-08-12

arxiv.org/abs/0808.1654v3

Topology and its Applications 156 (2009), 2659-2669

Mathematics

Combinatorics

Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Gr

Scientific paper

10.1016/j.topol.2009.04.014

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.

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