A Note on a Question of Erd\H{o}s & Graham

TL;DR

This paper investigates the partition regularity of the Pythagorean equation and related quadratic equations, providing a lower bound for the Rado number and positive results for similar equations.

Contribution

It offers the first known lower bound for the Rado number of the Pythagorean equation and extends results to two related quadratic equations.

Findings

Established a lower bound for the Rado number of x^2 + y^2 = z^2.

Proved positive results for the partition regularity of two similar quadratic equations.

Abstract

Erd\H{o}s & Graham ask whether the equation $x^2+y^2=z^2$ is partition regular, i.e. whether it has a finite Rado number. This note provides a lower bound and also states results in the affirmative for two similar quadratic equations.