On t-intersecting families of signed sets and permutations

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 de ne SX,k and S∗ X,k to be the families of k-signed r-sets given by SX,k := {{(x1, a1), ..., (xr, ar)} : a1, ..., ar are elements of {1, ..., k}}, S∗ X,k := {{(x1, a1), ..., (xr, ar)} : a1, ..., ar 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 de ne SF ,k := ⋃ F∈F SF,k and S∗ F ,k := ⋃ F∈F S∗ F,k. This paper features two theorems. The rst 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 rst one for S∗ F ,k.