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Monochromatic sums and products in ℕ
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Journal Article
Monochromatic sums and products in ℕ
2017
Overview
An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x + y,xy}. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of ℕ. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as x² – y² = z and x² + 2y² – 3z² = w.
Publisher
Department of Mathematics at Princeton University
