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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DOI: https://doi.org/10.1007/BF02122678