Stability for t-intersecting families of permutations

A family of permutations A subset of S(n) is said to be t-intersecting if any two permutations in A agree on at least t points i e for any sigma,pi epsilon A vertical bar{1 is an element of [n] sigma(i) = pi(i)}vertical bar >= 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 subset of S(n) has size at most (n 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 is an element of N and any positive constant c if A subset of S(n) is a t-intersecting family of permutations of size at least c(n-t)(1) 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 subset of 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 = {sigma epsilon Sn sigma(i) = 1 for all 1 <= t sigma(j)=j for some j > t+1} boolean OR {(1t+1), (2t+1), ,(t t + 1)}, which has size (1 - 1/e + o(1))(n-t)(1) Moreover if A is the same size as 1, then It must be a double translate of 7, meaning that there exist pi tau epsilon S(n) such that A = pi D tau 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) (C) 2010 Elsevier Inc All rights reserved