Monochromatic sums and products in $\mathbb{N}$

TL;DR

This paper proves that any finite coloring of natural numbers contains monochromatic solutions to certain non-linear equations, expanding the scope of partition regularity in Ramsey theory using topological dynamics.

Contribution

It introduces a new class of non-linear patterns that are guaranteed to appear monochromatically in any finite coloring of natural numbers, employing a novel correspondence principle.

Findings

Monochromatic solutions exist for equations like x^2 - y^2 = z in any finite coloring.

Established partition regularity for new non-linear equations.

Connected Ramsey theory with topological dynamics techniques.

Abstract

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 non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which 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 $x^2+2y^2-3z^2=w$.