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

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 $S(k)\ge 2^{2^{k-1}}$. The purpose of this note is to show that $S(k)\le 2^{2^{2^{O(k)}}}$.

6pp; for Endre Szemeredi's 80th Birthday volume

References in corpus (1)

• Monochromatic solutions to $x+y=z^2$