Exponential bounds for monochromatic sums equal to products

TL;DR

This paper proves that in any r-coloring of a large enough set of natural numbers, there exist monochromatic configurations where the sum of two numbers equals the product of two others, establishing exponential bounds.

Contribution

It introduces exponential bounds for the existence of monochromatic sum-product configurations in r-colored sets of natural numbers.

Findings

Existence of monochromatic sets with a+b=xy in large r-colored sets

Establishment of exponential size bounds for such sets

Extension of sum-product phenomena in combinatorics

Abstract

We show that any $r$-coloring of $\{1,...,r^{r^{r^{3r}}}\}$ contains monochromatic sets $\{a,b,a+b,x,y,xy\}$ with $a+b=xy.$