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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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