Stability for t-intersecting families of permutations

Abstract

A family of permutations A ⊂ S n is said to be t-intersecting if any two permutations in A agree on at least t points, i.e. for any σ , π ∈ A , | { i ∈ [ n ] : σ ( i ) = π ( i ) } | ⩾ t . It was proved by Friedgut, Pilpel and the author in [6] that for n sufficiently large depending on t, a t-intersecting family A ⊂ S n has size at most ( n − t ) ! , with equality only if A is a coset of the stabilizer of t points (or ‘ t-coset’ for short), proving a conjecture of Deza and Frankl. Here, we first obtain a rough stability result for t-intersecting families of permutations, namely that for any t ∈ N and any positive constant c, if A ⊂ S n is a t-intersecting family of permutations of size at least c ( n − t ) ! , then there exists a t-coset containing all but at most an O ( 1 / n ) -fraction of A . We use this to prove an exact stability result: for n sufficiently large depending on t, if A ⊂ S n is a t-intersecting family which is not contained within a t-coset, then A is at most as large as the family D = { σ ∈ S n : σ ( i ) = i ∀ i ⩽ t , σ ( j ) = j for some j > t + 1 } ∪ { ( 1 t + 1 ) , ( 2 t + 1 ) , … , ( t t + 1 ) } , which has size ( 1 − 1 / e + o ( 1 ) ) ( n − t ) ! . Moreover, if A is the same size as D then it must be a ‘double translate’ of D , meaning that there exist π , τ ∈ S n such that A = π D τ . The t = 1 case of this was a conjecture of Cameron and Ku and was proved by the author in [5]. We build on our methods in [5], but the representation theory of S n and the combinatorial arguments are more involved. We also obtain an analogous result for t-intersecting families in the alternating group A n .