Representation of integers as monochromatic sums of squares of primes

For any integer K≥1, let s(K) be the smallest integer such that when the set of squares of the prime numbers is coloured in K colours, each sufficiently large integer can be written as a sum of no more than s(K) squares of primes, all of the same colour. We show that s(K)≪Kexp⁡((3log⁡2+o(1))log⁡Klog⁡log⁡K) for K≥2. This upper bound for s(K) is close to optimal and improves on s(K)≪ϵK2+ϵ, which is the best available upper bound for...