A chromatic version of Lagrange’s four squares theorem

Abstract

When the sequence of squares of the natural numbers is coloured with $K$ colours, $K\ge 1$ an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as the sum of no more than $s(K)$ squares, all of the same colour. We show that $s(K) \ll _{\epsilon } K^{2 + \epsilon }$, for any $\epsilon > 0$ and all $K \ge 1$, improving on a result of Hegyvári and Hennecart, who obtained $s(K) \ll (K\log K)^5$.