Around the Complete Intersection Theorem

Around the Complete Intersection Theorem

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)...

The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets

The theorem presented and proved in this paper can be viewed as an extension or improvement of our recent Complete Intersection Theorem [1] and may be called the Complete Nontrivial-Intersection Theorem. It goes considerably beyond the well-known Hilton Milner Theorem [10]. We put the result into the proper perspective with a brief sketch of the key steps in its development, beginning with the ...

The Complete Intersection Theorem for Systems of Finite Sets

We are concerned here with one of the oldest problems in combinatorial extremal theory. It is readily described after we have made a few conventions. ‫ގ‬ denotes the set of

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...