MathNet 2013 Question 52

Let $A_{1}, A_{2}, \ldots, A_{m}$ be finite sets of size $2012$ and let $B_{1}, B_{2}, \ldots, B_{m}$ be finite sets of size $2013$ such that $A_{i} \cap B_{j} = \emptyset$ if and only if $i = j$. Find the maximum value of $m$.