Assume $m$, $n$ to be positive integers. Given two partitions $[mn]=A_1 \cup A_2 \cup \dots \cup A_m$ and $[mn]=B_1 \cup B_2 \cup \dots \cup B_m$ of $[mn]$ into sets of cardinality $n$, show that the sets $A_i$ can be renumbered in a way s.t. $A_i \cap B_i \ne \emptyset$ for all $i \in [m]$.
I am trying to construct a connected/associated bipartite graph such that each set $A_i$ or $B_i$ is a vertex and the edge represent the intersection of two sets/vertices (and maybe utilize Hall's theorem.)
But I am having a bit of trouble to construct such a graph and am not sure if "$A_i \cap B_i \ne \emptyset$ for all $i \in [m]$" in the question means the same $i$ or the $i$'s can be different.