Let FC(),n≥2k. -1 S,T∈F,SnT卡0|F≤ k-1 when n>2k,induction on n WLOG:F is shifted F1={S∈F|n∈S}F1={S\{n}|S∈F} Fis intersecting otherwise, 3A,B∈F A0B={n} |AUB|≤2k-1<n-1> Ji<n,ig AUB C=A\{n}U{}∈F☐ F is shifted C∩B=0 contradiction!Let F ￾ ￾[n] k ⇥ , n ⇥ 2k. |F| ⇥ ￾n ￾ 1 k ￾ 1 ⇥ ⇤S, T ￾ F, S ⌃ T ⇥= ⌅ when n > 2k, induction on n F1 = {S ￾ F | n ￾ S} WLOG: F is shifted F￾ 1 = {S \ {n} | S ￾ F1} is intersecting otherwise, < n ￾ 1 ￾ F = ￾ ⇥A, B ￾ F A ￾ B = {n} |A ⇤ B| ⇥ 2k ￾ 1 C = A \ {n} ￾ {i} C ￾ B ⇤i < n, i ⇥￾ A ⌅ B F is shifted contradiction! F￾ 1