The weighted complete intersection theorem

摘要

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 mu(p), given by mu(p)(A) = p(|A|) (1 - p)(n-|A|). Our theorem gives the maximum pt, measure of a t-intersecting family on n points, and describes all optimal families for t >= 2. (C) 2017 Elsevier Inc. All rights reserved.