Intersecting families of permutations

Abstract

A set of permutations I ⊂ S n I \subset S_n is said to be k k -intersecting if any two permutations in I I agree on at least k k points. We show that for any k ∈ N k \in \mathbb {N} , if n n is sufficiently large depending on k k , then the largest k k -intersecting subsets of S n S_n are cosets of stabilizers of k k points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning k k -cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.