Monochromatic solutions to $x+y=z^2$ in the interval $[N,cN^4]$

TL;DR

This paper improves the known bounds for the existence of monochromatic solutions to the equation x+y=z^2 in 2-colored natural numbers, reducing the interval size needed for guaranteed solutions.

Contribution

It provides a new proof of Green and Lindqvist's theorem and tightens the interval bounds for monochromatic solutions in 2-colorings.

Findings

Monochromatic solutions exist in every interval [N, 10^4 N^4] for large N.

A 2-coloring avoiding solutions is constructed in [N, (1/27)N^4], showing bounds are tight up to a constant factor.

The proof simplifies previous methods and improves the interval bounds for solutions.

Abstract

Green and Lindqvist proved that for any 2-colouring of $\mathbb{N}$, there are in\-fi\-ni\-tely many monochromatic solutions to $x+y=z^2$. In fact, they showed the existence of a monochromatic solution in every interval $[N,cN^8]$ with large enough $N$. In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval $[N,10^4N^4]$ with large enough $N$. A 2-colouring of $[N,(1/27)N^4]$ avoiding monochromatic solutions to $x+y=z^2$ is also presented, which shows that in $10^4N^4$ only the constant factor can be reduced.