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 ₂g] sequence $ \mathcal{A} $ \mathcal{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 Erdős and Renyi in 1960. We obtain a similar result for B ₂g] sequences of squares.