An extremal problem for sets: A new approach via Bezoutians

Abstract

By means of Bezoutians, we give a new and short proof of Alon's theorem: Let X1, ..., Xn be n disjoint sets. For 1 ≤ i ≤ n and 1 ≤ j ≤ h let Aij and Bij be subsets of Xi that satisfy |Aij| ≤ ri and |Bij| ≤ si for 1 ≤ i ≤ n, 1 ≤ j ≤ h, ({n-ary union}iAij)∩({n-ary union}iBij)=φ for 1 ≤ j ≤ h, ({n-ary union}iAij)∩({n-ary union}iBil)≠φ for 1 ≤ j ≤ l ≤ h, then h≤ ∏ i=1 n ( ri+si ri.. © 1993.