Lemma(Sperner) Let F().Then VF1≥ n-k +iF列 (for k<n) k △F1≥ (for ke Corollary: Ifk≤(n-1),then VF|≥F. Ifk≥(m-1),then|△F|≥F.Let F ￾ ￾[n] k ⇥ . Then |⇧F| ⇥ n ￾ k k + 1 |F| |￾F| ⇥ k n ￾ k + 1|F| (for k<n) (for k > 0) Let F ￾ ￾[n] k ⇥ . Then |⇧F| ⇥ n ￾ k k + 1 |F| |￾F| ⇥ k n ￾ k + 1|F| (for k<n) (for k > 0) Lemma (Sperner) Corollary: If k ⇥ 1 2 (n ￾ 1), then |⌃F| ⇤ |F|. If k ⇥ 1 2 (n ￾ 1), then |￾F| ⇥ |F|