The Forbidden Cross Intersection Problem for Permutations

We prove the following, for a universal constant $c>0$. Let $n \in \mathbb{N}$ and $1 \leq t<c\frac{n}{\log n}$. Let $F,G \subset S_n$ be families of permutations such that no $\sigma \in F$ and $\tau \in G$ agree on exactly $t-1$ values. Then $|F||G| \leq ((n-t)!)^2$, with equality if and only if $F=G=\{\sigma \in S_n:\sigma(i_1)=j_1,\ldots,\sigma(i_t)=j_t\}$, for some $i_1,\ldots,i_t,j_1,\ldots,j_t \in \{1,2,\ldots,n\}$. The range of values of $t$ in the result is essentially optimal, as for any $\epsilon>0$, the statement fails for $t=(1+\epsilon)\frac{n}{\log_2 n}$ and all $n>n_0(\epsilon)$. This solves the cross-intersection variant of the Erd\H{o}s-S\'{o}s forbidden intersection problem for permutations. The best previously known result, by Kupavskii and Zakharov (Adv.~Math., 2024), obtained the same assertion for $t \leq \tilde{O}(n^{1/3})$. We obtain our result by combining two recently introduced techniques: hypercontractivity of global functions and spreadness.