Setwise intersecting families of permutations

A family of permutations A ⊂ S n is said to be t-set-intersecting if for any two permutations σ , π ∈ A , there exists a t-set x whose image is the same under both permutations, i.e. σ ( x ) = π ( x ) . We prove that if n is sufficiently large depending on t, the maximum-sized t-set-intersecting families of permutations in S n are cosets of stabilizers of t-sets. The t = 2 case of this was conjectured by János Körner. It can be seen as a variant of the Deza–Frankl conjecture, proved in Ellis, Friedgut and Pilpel (2011) [3]. Our proof uses similar techniques to those of Ellis, Friedgut and Pilpel (2011) [3], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.