Sperner's Theorem Fc2 is an antichain.Then F≤(ny2） Lubell's proof(double counting) >lS1(n-IS)I≤nd S∈F IS(m-1S1)！ n! ≤1 S∈F F1≤(2〉Sperner’s Theorem F ￾ 2[n] is an antichain. Then |F| ⇥ ￾ n ￾n/2⇥ ⇥ . |F| ￾ n ￾n/2⇥ ⇥ ￾ = ￾ 1 |F| ￾ ￾ n ￾n/2⇥ ⇥ Lubell’s proof (double counting) ￾ S￾F |S|!(n ￾ |S|)! ⇥ n! ⇤ S￾F 1 ￾ n |S| ⇥ ￾ S￾F |S|!(n ￾ |S|)! n!