An Asymptotic Complete Intersection Theorem for Chain Products

An Asymptotic Complete Intersection Theorem for Chain Products

For positive integers k, ` and n let N`(n, k) = {a = (a1, . . . , an) : ai ∈ {0, 1, . . . , k}, i = 1, . . . , n,∑ ai = `}. A family F ⊆ N`(n, k) is called t-intersecting if for all a, b ∈ F there exist t coordinates i1, . . . , it such that ai j , bi j ≥ 1 holds for j = 1, . . . , t . Define M`(n, k, t) = max{|F | : F ⊆ N`(n, k),F is t-intersecting}. N`(n, k) can be viewed as the `-th level of...

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)...

Around the Complete Intersection Theorem

The Intersection Theorem for Direct Products

An intersection theorem for supermatroids

We generalize the matroid intersection theorem to distributive supermatroids, a structure that extends the matroid to the partially ordered ground set. Distributive supermatroids are special cases of both supermatroids and greedoids, and they generalize polymatroids. This is the first good characterization proved for the intersection problem of an independence system where the ground set is par...