Monochromatic products and sums in the rationals

TL;DR

This paper proves that in any finite coloring of the rational numbers, there exists a monochromatic set containing two numbers, their sum, and their product, demonstrating a specific combinatorial structure.

Contribution

It establishes a new combinatorial result about monochromatic configurations involving sums and products in the rationals.

Findings

Existence of monochromatic sets with sum and product in any finite coloring of rationals

Extension of classical Ramsey theory to rational numbers

Identification of specific algebraic structures within colored rationals

Abstract

We show that for every coloring of the rationals into finitely many colors, one of the colors contains a set of the form $\{x,y,xy,x+y\}$ for some nonzero $x$ and $y$.