Problem 3. Let k, l be two fixed positive integers. Let {(Ai , Bi), 1 i h} be a family of pairs of subsets of the set of integers such that: |Ai | = k for all i, |Bi | = l for all i, Ai Bi = for all i, and (Ai Bj ) (Aj Bi) 6= for all i 6= j. Prove that h ((k+l)^k+l)/k^k(l^l) . (Hint: consider coloring the numbers red or blue with probability p)