On monochromatic sums of squares of primes

Abstract Let K be a positive integer, { A i , 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 A i . In this paper, we prove that s ( K ) ≪ ϵ K 2 + ϵ for sufficiently small positive number ϵ and all K ≥ 1 .