Monochromatic sums of squares.

For any integer K≥1<inline-graphic></inline-graphic> 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)≥Kexp(log2+o(1))logKloglogK<inline-graphic></inline-graphic>. In this article we show that s(K)≤Kexp(3log2+o(1))logKloglogK<inline-graphic></inline-graphic>. This improves on the bound s(K)≪ϵK2+ϵ<inline-graphic></inline-graphic>, which is the best available upper bound for s(K). [ABSTRACT FROM AUTHOR]