Setwise intersecting families of permutations

A family of permutations $A \subset S_n$ is said to be \emph{$t$-set-intersecting} if for any two permutations $\sigma, \pi \in A$, there exists a $t$-set $x$ whose image is the same under both permutations, i.e. $\sigma(x)=\pi(x)$. We prove that if $n$ is sufficiently large depending on $t$, the largest $t$-set-intersecting families of permutations in $S_n$ are cosets of stabilizers of $t$-sets. The $t=2$ case of this was conjectured by J\'anos K\"orner. It can be seen as a variant of the Deza-Frankl conjecture, proved in [4]. Our proof uses similar techniques to those of [4], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.