A Note on P-sets

A P-set is a set S of positive integers such that no element of S divides the sum of any two (not necessarily being different) larger elements. Erdös and Sárközy [2] conjectured that there exists a constant c > 0 such that for every P-set S we have |{s ∈ S : s ≤ N}| < N1−c for infinitely many integers N . For P-sets S consisting of pairwise coprime integers this conjecture has been proved by T. Schoen [6] who showed that in this case we have |{s ∈ S : s ≤ N}| < 2N for infinitely many integers N . In the present note we prove that the term 2N in Schoen′s estimate can be replaced by (3+ε)N/ logN . Our method uses the arithmetic form of the large sieve as well as mean value estimates for multiplicative functions. (Mathematics Subject Classification (2000): 11B05, 10H25)