The weighted complete intersection theorem

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families for t ≥ 2. We extend this theorem to the weighted setting, in which we consider unconstrained families. The goal in this setting is to maximize the μp measure of the family, where the measure μp is given by μp(A) = p |A|(1− p)n−|A|. Our theorem gives the maximum μp measure of a t-intersecting family on n points, and describes all optimal families for t ≥ 2.