Upper and lower bounds on $B_k^+$-sets

Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a $B_k^+$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\{1, 2, \dots, N \} \subset \integers$ or in the cyclic group $\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^+$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^+$-set contained in the interval $\{1, 2, \dots, N \}$. For odd $k \geq 3$, we construct $B_k^+$-sets that have more elements than the $B_k$-sets constructed by Bose and Chowla. We prove a $B_3^+$-set $A \subset \integers_N$ has at most $(1 + o(1))(8N)^{1/3}$ elements. Finally we obtain new upper bounds on the maximum size of a $B_k^*$-set $A \subset \{1,2, \dots, N \}$, a problem first investigated by Ruzsa.