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)...

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...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

We consider quantitative versions of Helly-type questions, that is, instead finding a point in the intersection, we bound volume intersection. Our first main result is version Fractional Helly Theorem Katchalski and Liu, second one ( p , q ) -Theorem Alon Kleitman.

We introduce the following generalization of set intersection via characteristic vectors: for n,q,s,t≥1 a family F⊆{0,1,…,q}n vectors is said to be s-sum t-intersecting if any distinct x,y∈F there exist at least t coordinates, where entries x and y sum up s, i.e. |{i:xi+yi≥s}|≥t. The original corresponds case q=1,s=2. address analogs several variants classical results in this setting: Erdős–Ko–...

A fixed-point theorem on compact compositions of acyclic maps on admissible (in the sense of Klee) convex subset of a t.v.s. is applied to obtain a cyclic coincidence theorem for acyclic maps, generalized “onNeumann type intersection theorems, the Nash type equilibrium theorems, and the “onNeumann minimax theorem. Our new results generalize earlier works of Lassonde [l], Simons [2], and Park [3...

We propose new intersection problems in the q-ary n-dimensional hypercube. The answers to the problems include the Katona’s t-intersection theorem and the Erdős–Ko–Rado theorem as special cases. We solve some of the basic cases of our problems, and for example we get an Erdős–Ko–Rado type result for t-intersecting k-uniform families of multisets with bounded repetitions. Another example is obta...