Lemma(Sperner) LetF().Then n-k F列≥太+i列 (for k <n) n(for k △F1≥ double counting R={(S,T)S∈F,T∈VF,ScT} VS∈F, n-kT∈(r)have TS 1R=(n-k)儿F到 VTVF,T has ()+1 many k-subsets |R≤(k+1)川VFLet 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) double counting ⇥S ￾ F, n ￾ k T ⇤ ￾ [n] k+1⇥ have T ⇥ S R = {(S, T) | S ⇥ F, T ⇥ ￾F, S ￾ T} |R| = (n ￾ k)|F| ⇥T ￾ ⌅F, T has ￾k+1 k ⇥ = k + 1 many k-subsets |R| ￾ (k + 1)|⇧F|