Abstract
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.