Monochromatic sums of squares

Abstract

For any integer $K\ge 1$ let s(K) be the smallest integer such that in any colouring of the set of squares of the integers in K colours every large enough integer can be written as a sum of no more than s(K) squares, all of the same colour. A problem proposed by Sárközy asks for optimal bounds for s(K) in terms of K. It is known by a result of Hegyvári and Hennecart that $s(K) \ge K \exp \left( \frac{(\log 2 + \mathrm{o}(1))\log K}{\log \log K}\right)$. In this article we show that $s(K) \le K \exp \left( \frac{(3\log 2 + \mathrm{o}(1))\log K}{\log \log K}\right)$. This improves on the bound $s(K) \ll _{\epsilon } K^{2 +\epsilon }$, which is the best available upper bound for s(K).