On monochromatic solutions to $x-y=z^2$

TL;DR

This paper investigates the bounds on the largest number for which a k-coloring avoids monochromatic solutions to the equation x - y = z^2, establishing an upper bound that is triple-exponential in k.

Contribution

It provides an improved upper bound on S(k), showing that it grows at most triple-exponentially with respect to the number of colors k.

Findings

Established that S(k) is at most double-exponential in a double-exponential function of k.

Confirmed the existence of S(k) for all k via prior results.

Provided a new upper bound improving previous estimates.

Abstract

For $k \in \mathbb{N}$, write $S(k)$ for the largest natural number such that there is a $k$-colouring of $\{1,\dots,S(k)\}$ with no monochromatic solution to $x-y=z^2$. That $S(k)$ exists is a result of Bergelson, and a simple example shows that $S(k) \geq 2^{2^{k-1}}$. The purpose of this note is to show that $S(k)\leq 2^{2^{2^{O(k)}}}$.