On monochromatic solutions to x−y=z2

For k∈N, write S(k) for the largest natural number such that there is a k-colouring of {1,…,S(k)} with no monochromatic solution to x−y=z2. That S(k) exists is a result of Bergelson, and a simple example shows that S(k)≥22k−1. The purpose of this note is to show that S(k)≤222O(k) .