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 multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots ,n\}$. In this paper we improve the best-known upper bounds on $F_{k,g}(n)$ 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 $F_{k,g}(n)$ that matches a construction of Lindstr"om while removing one of the hypotheses.