Monochromatic sums and quotients in $\mathbb N$

TL;DR

This paper proves a strong form of partition regularity for specific numerical configurations involving sums and quotients, extending classical theorems like Hindman's.

Contribution

It establishes a new infinitary partition regularity result for configurations including sums and quotients, and reduces complex cases to simpler ones.

Findings

Proves partition regularity of x,y,x+y,y/x in an infinitary setting.

Extends Hindman's Theorem to new configurations.

Reduces the problem of partition regularity involving polynomial products to simpler cases.

Abstract

We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.