Minimum pair of integer sets with distinct sum of pairs
The bound is exactly $mn+1$: one can take $$ A = \{1,2,\dots,m\} \quad\text{and}\quad B = \{1,m+1,2m+1,\dots,m(n-1)+1\} $$ to achieve $mn+1$ for the maximal sum, and we can't do any better because all $mn$ sums are integers greater than or equal to $2$.