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 | {n-ary intersection}_{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 Erdo{combining double acute accent}s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].