Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(ϵ) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x1/h-ϵ.
We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies gh(ϵ) ≪ ϵ−1, improving the bound gh(ϵ) ≪ ϵ−h+1 obtained by Vu.
Finally we use the “alteration method” to get a better bound for g3(ϵ), obtaining a more precise estimate for the growth of B3[g] sequences. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010