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 on n points with respect to the measure μp 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.