Monochromatic sums of squares
Résumé
For any integer 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 Sarkozy asks for optimal bounds for s(K) in terms of K. It is known by a result of Hegyvari and Hennecart that . In this article we show that . This improves on the bound , which is the best available upper bound for s(K).