A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121–138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150–153]. Let 2 ⩽ t + 1 ⩽ k ⩽ 2 t + 1 and n ⩾ ( t + 1 ) ( k − t + 1 ) . Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If | ⋂ F ∈ F F | < t , then | F | ⩽ ( t + 2 ) ( n − t − 2 k − t − 1 ) + ( n − t − 2 k − t − 2 ) , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121–138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erdős–Ko–Rado theorem, Combinatorica 4 (1984) 247–257].