Upper and lower bounds on the size of B-k[g] sets

A subset A of the integers is a B-k[g] set if the number of k-element multisets from A that sum to any fixed integer is at most g. Let F-k,F-g(n) denote the maximum size of a B-k[g] set in {1, ..., n}. In this paper we improve the best-known upper bounds on F-k,F-g(n) for g > 1 and k large. When g =1 we match the best upper bound of Green with an improved error term. Additionally, we give a lower bound on F-k,F-g(n) that matches a construction of Lindstrom while removing one of the hypotheses.