17 2.The Probabilistic Method others,the famous Sperner theorem:If F is a family of subsets of [m]with no 。 17 2. The Probabilistic Method others, the famous Sperner theorem: If F is a family of subsets of [m] with no two distinct sets A, B ∈ F satisfying A ⊂ B, then |F| ≤ ￾ m ⌊m/2⌋
. To see this, set F = {A1, A2, . . . , An} and Bi = [m] \ Ai , and use the fact that ￾m k
≤ ￾ m ⌊m/2⌋
for all k = 0, 1, . . . , m