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Monochromatic sums and products in \(\mathbb{N}\). (English) Zbl 1430.05128
Summary: 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 \(\mathbb{N}\). 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 \(x^2-y^2=z\) and \(x^2+2y^2-3z^2=w\).

MSC:
05D10 Ramsey theory
11B75 Other combinatorial number theory
37A30 Ergodic theorems, spectral theory, Markov operators
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