Probabilistic constructions of B 2[g] sequences

We use the probabilistic method to prove that for any positive integer g there exists an infinite B 2[g] sequence A = {a k } such that a k . k 2+1/g(log k)1/g+o(1) as k... The exponent 2+1/g improves the previous one, 2 + 2/g, obtained by Erd.s and Renyi in 1960. We obtain a similar result for B 2[g] sequences of squares.