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-signedr-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"@?"FS"F","k and S"F","k^*@?@?"F"@?"FS"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^*.
['Peter Borg']
On t-intersecting families of signed sets and permutations