Monochromatic products and sums in $2$-colorings of $\mathbb{N}$

TL;DR

This paper proves that in any 2-coloring of natural numbers, there are infinitely many monochromatic sets involving sums and products, extending classical theorems like Hindman's and introducing new monochromatic configurations.

Contribution

It introduces new monochromatic set configurations involving sums and products, extending Hindman's theorem and providing a colorful variant for balanced colorings.

Findings

Existence of infinitely many monochromatic sets of the form {x, y, xy, x+y}

Generalization to sets involving multiple variables and sums/products

Extension of Hindman's theorem to product-sum configurations

Abstract

We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any 'balanced' coloring.