Large non-trivial $t$-intersecting families for signed sets

Abstract

For positive integers n, r, k with n r and k 2, a set {(x 1 , y 1 ), (x 2 , y 2 ), . . . , (x r , y r )} is called a k-signed r-set on [n] if x 1 , . . . , x r are distinct elements of [n] and y 1 . . . , y r ∈ [k]. We say a t-intersecting family consisting of k-signed r-sets on [n] is trivial if each member of this family contains a fixed k-signed t-set. In this paper, we determine the structure of large maximal non-trivial t-intersecting families. In particular, we characterize the non-trivial t-intersecting families with maximum size for t 2, extending a Hilton-Milner-type result for signed sets given by Borg.