Let Sn denote the set of permutations of [n]={1,2,…,n}. For a positive integer k, define Sn,k to be the set of all permutations of [n] with exactly k disjoint cycles, i.e.,
Sn,k={π∈Sn:π=c1c2⋯ck},
where c1,c2,…,ck are disjoint cycles. The size of Sn,k is given by , where s(n,k) is the Stirling number of the first kind. A family A⊆Sn,k is said to be 3342e6391ef65bfd1b03" title="Click to view the MathML source">t-cycle-intersecting if any two elements of A have at least 3342e6391ef65bfd1b03" title="Click to view the MathML source">t common cycles. A family A⊆Sn,k is said to be trivially 3342e6391ef65bfd1b03" title="Click to view the MathML source">t-cycle-intersecting if A is the stabiliser of 3342e6391ef65bfd1b03" title="Click to view the MathML source">t fixed points, i.e., A consists of all permutations in Sn,k with some 3342e6391ef65bfd1b03" title="Click to view the MathML source">t fixed cycles of length one. For 1≤j≤t, let
For t+1≤s≤k, let
In this paper, we show that, given any positive integers k,t with k≥2t+3, there exists an n0=n0(k,t), such that for all n≥n0, if A⊆Sn,k is non-trivially 3342e6391ef65bfd1b03" title="Click to view the MathML source">t-cycle-intersecting, then
|A|≤|B|,
where . Furthermore, equality holds if and only if A is a conjugate of B, i.e., A=β−1Bβ for some β∈Sn.