  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.
