Bounds for monochromatic solutions to $\{x+y,xy\}$

TL;DR

This paper establishes bounds on the size of the set [N] such that any r-coloring guarantees a monochromatic solution to the set {x+y, xy} with x > y > 2, highlighting combinatorial properties of these equations.

Contribution

The paper provides explicit bounds for N in terms of r, ensuring the existence of monochromatic solutions to {x+y, xy} in r-colored sets, advancing understanding of combinatorial number theory.

Findings

Any r-coloring of [N] with N ≥ expexp(r^{50}) contains a monochromatic {x+y, xy} with x > y > 2.

The bounds are double-exponential in r, indicating the complexity of avoiding monochromatic solutions.

The results extend previous work on monochromatic solutions to additive and multiplicative equations.

Abstract

Let $r$ be a sufficiently large positive integer, and let $N \ge \exp\exp(r^{50})$. Then any $r$-colouring of $[N]$ contains a monochromatic copy of $\{x+y,xy\}$ with $x > y > 2$.