Monochromatic sums and products in \(\mathbb{N}\). (English) Zbl 1430.05128

Moreira, Joel

Ann. Math. (2) 185, No. 3, 1069-1090 (2017).

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

Keywords:

Ramsey theory; partition regularity; polynomial configurations