Probabilistic Methods in Combinatorics

Remark. This is also known as the Lubell-Yamamoto-Meshalkin inequality. Solution: Generate a random permutation σ = (x1, . . . , xM ) of [M ]. Let each Ei be the event that the set Ai appears as the initial segment of σ, i.e., that {x1, . . . , xai} = Ai. Since none of the Ai contain each other, the Ei are mutually exclusive. But each P [Ei] = ( M ai )−1 , so the LHS is P [∪Ei], and probabilities are always ≤ 1. 2. (Very slight simplification of a problem in the MathLinks.ro 2008 contest) LetA1, . . . , An andB1, . . . , Bn be distinct finite subsets of N such that: • for every i, Ai ∩Bi = ∅, and • for every i 6= j, (Ai ∩Bj) ∪ (Aj ∩Bi) 6= ∅. Prove that for every real number 0 ≤ p ≤ 1, ∑