Let $A, B \in Z^+$ be two sets where $|A|=m, |B|=n$. If all pairwise sums $a+b (a\in A, b\in B)$ are distinct, a.k.a. $|A+B|=|A||B|$, what would be the minimum value of $max(A\cup B)$?
A trivial bound would be $mn/2$ as the largest pairwise sum is at least $mn$. Meanwhile constructively we can create 2 sets $A=\{1, 2, \ldots, m\}, B=\{1, m+1, 2m+1, \ldots, (n-1)m+1\}$ that have distinct pairwise sums. So $mn/2 \leq min\{max(A\cup B)\} \leq mn-\max\{n, m\}+1$. What would be a better bound for this problem?