Ultrafilters and multidimensional Ramsey Theorems

Abstract

Utilizing ultrafilters on the setN of natural numbers which have certain special properties, we prove some simultaneous generalizations of Ramsey's Theorem and several single dimension Ramsey-type theorems. (For instance, as a very special case, we obtain a simultaneous generalization of Ramsey's Theorem and van der Waerden's Theorem.) Examples are given to show directions in which these results cannot be extended.

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Bergelson, V., Hindman, N. Ultrafilters and multidimensional Ramsey Theorems. Combinatorica 9, 1–7 (1989). https://doi.org/10.1007/BF02122678

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AMS subject classification (1980)

  • 05A17