A proof of the Cameron-Ku conjecture

A family of permutations A ⊂ Sn is said to be intersecting if any two permutations in A agree at some point, i.e. for any σ, π ∈ A, there is some i such that σ(i) = π(i). Deza and Frankl [3] showed that for such a family, |A| ≤ (n− 1)!. Cameron and Ku [2] showed that if equality holds then A = {σ ∈ Sn : σ(i) = j} for some i and j. They conjectured a ‘stability’ version of this result, namely that there exists a constant c < 1 such that if A ⊂ Sn is an intersecting family of size at least c(n− 1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family ‘centred’). They also made the stronger ‘Hilton-Milner’ type conjecture that for n ≥ 6, if A ⊂ Sn is a non-centred intersecting family, then A cannot be larger than the family C = {σ ∈ Sn : σ(1) = 1, σ(i) = i for some i > 2}∪{(12)}, which has size (1− 1/e+ o(1))(n − 1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of Sn. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n ≥ 4, if A,B ⊂ Sn are cross-intersecting, then |A||B| ≤ ((n − 1)!). This was a conjecture of Leader [11]; it was proved for n sufficiently large by Friedgut, Pilpel and the author in [4].