On t-intersecting families of signed sets and permutations

Abstract

A family A of sets is said to be t-intersecting if any two sets in A contain at least t common elements. A t-intersecting family is said to be trivial if there are at least t elements common to all its sets.Let X be an r-set {x1,…,xr}. For k≥2, we define SX,k and to be the families of k-signedr-sets given by SX,k≔{{(x1,a1),…,(xr,ar)}:a1,…,ar are elements of {1,…,k}}, can be interpreted as the family of permutations of r-subsets of {1,…,k}. For a family F, we define SF,k≔⋃F∈FSF,k and .This paper features two theorems. The first one is as follows: For any two integers s and t with t≤s, there exists an integer k0(s,t) such that, for any k≥k0(s,t) and any family F with t≤max{|F|:F∈F}≤s, the largest t-intersecting sub-families of SF,k are trivial. The second theorem is an analogue of the first one for .