Forbidding just one intersection, for permutations

Abstract

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of 2 points. We also obtain a Hilton-Milner type result, namely that if $\mathcal{A}$ is such a family which is not contained within a coset of the stabilizer of 2 points, then it is no larger than the family $\{\sigma \in S_{n}:\ \sigma(1)=1,\sigma(2)=2,\ \#\{\textrm{fixed points of}\sigma \geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\ 4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}$. We conjecture that for $t \in \mathbb{N}$, and for $n$ sufficiently large depending on $t$, if $\mathcal{A}$ is family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly $t-1$ times, then $|\mathcal{A}| \leq (n-t)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of $t$ points. This can be seen as a permutation analogue of a conjecture of Erd\H{o}s on families of $k$-element sets with a forbidden intersection, proved by Frankl and F\"uredi in [P. Frankl and Z. F\"uredi, Forbidding Just One Intersection, Journal of Combinatorial Theory, Series A, Volume 39 (1985), pp. 160-176].