On t-intersecting families of permutations

We prove that there exists a constant c0 such that for any t∈N and any n≥c0t, if A⊂Sn is a t-intersecting family of permutations then |A|≤(n−t)!. Furthermore, if |A|≥0.75(n−t)! then there exist i1,…,it and j1,…,jt such that σ(i1)=j1,…,σ(it)=jt holds for any σ∈A. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on t-intersecting families of permutations hold for all t≤c0n. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to t-intersecting sub-families of ‘pseudorandom’ families in {1,2,…,n}n, like Sn.