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A subset $A$ of the integers is a $B_k[g]$ set if the number of multisetsfrom $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denotethe maximum size of a $B_k[g]$ set in $\{1,\dots, n\}$. In this paper weimprove 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 constructionof Lindstr\"om while removing one of the hypotheses.

Upper and lower bounds on the size of $B_k[g]$ sets