A Complete Intersection Theorem for Large Permutation Groups

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements.

We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size of a $t$-intersecting family in $S_n$ is obtained by one of the families $\mathcal{F}_{n,t,r}=\{\sigma \in S_n: |\mathrm{Fixed}(\sigma) \cap \{1,2,\ldots,t+2r\}|\geq t+r\}$, where $\mathrm{Fixed}(\sigma)$ is the set of fixed points of $\sigma$.

This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977).