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) << K exp ((3 log 2+o(1) log K/log log K) for K >= 2. This upper bound for s(K) is close to optimal and improves on s(K) <<(epsilon) K2+epsilon, which is the best available upper bound for s(K). (C) 2020 Elsevier Inc. All rights reserved.