On monochromatic sums of squares of primes

Let K be a positive integer, {Ai,1≤i≤K} be any partition of the sequence of squares of primes and s(K) be the smallest positive integer such that every sufficiently large integer can be written as the sum of no more than s(K) elements, which belong to one of the sets Ai. In this paper, we prove that s(K)≪ϵK2+ϵ for sufficiently small positive number ϵ and all K≥1.