On [formula omitted]-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 { x 1 , … , x r } . For k ≥ 2 , we define S X , k and S X , k ∗ to be the families of k - signed r - sets given by S X , k ≔ { { ( x 1 , a 1 ) , … , ( x r , a r ) } : a 1 , … , a r are elements of { 1 , … , k } } , S X , k ∗ ≔ { { ( x 1 , a 1 ) , … , ( x r , a r ) } : a 1 , … , a r are distinct elements of { 1 , … , k } } . S X , k ∗ can be interpreted as the family of permutations of r -subsets of { 1 , … , k } . For a family F , we define S F , k ≔ ⋃ F ∈ F S F , k and S F , k ∗ ≔ ⋃ F ∈ F S F , k ∗ . 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 k 0 ( s , t ) such that, for any k ≥ k 0 ( s , t ) and any family F with t ≤ max { | F | : F ∈ F } ≤ s , the largest t -intersecting sub-families of S F , k are trivial. The second theorem is an analogue of the first one for S F , k ∗ .