LAUSR

For any integer $$K\ge 1$$K≥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 $$s(K) \ge K \exp \left( \frac{(\log 2 + \mathrm{o}(1))\log K}{\log \log K}\right) $$s(K)≥Kexp(log2+o(1))logKloglogK. In this article we show that $$s(K) \le K \exp \left( \frac{(3\log 2 + \mathrm{o}(1))\log K}{\log \log K}\right) $$s(K)≤Kexp(3log2+o(1))logKloglogK. This improves on the bound $$s(K) \ll _{\epsilon } K^{2 +\epsilon }$$s(K)≪ϵK2+ϵ, which is the best available upper bound for s(K).