Sharp stability for cross $t$-intersecting families of permutations in the linear range

We obtain several intersection theorems for permutations with purely combinatorial proofs. Two families $\mathcal{F},\mathcal{G}\subseteq S_n$ are cross $t$-intersecting if every $\sigma \in \mathcal{F}$ and $\tau \in \mathcal{G}$ agree on at least $t$ points. A $t$-coset is a coset of the stabilizer of $t$ points. A subset of $S_n$ is non-trivial if it is not contained in any $t$-coset. Let $d_m$ denote the $m$-th derangement number. We prove that, for all $t\ge 1$ and $n\ge 400t$, every pair of cross $t$-intersecting families $\mathcal{F},\mathcal{G}\subseteq S_n$ satisfies the following: (i) $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le ((n-t)!-d_{n-t}-d_{n-t-1})((n-t)!+t)$ if $\mathcal{F}\cup \mathcal{G}$ is non-trivial. (ii) $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le ((n-t)!-d_{n-t}-d_{n-t-1}+t{)}^2$ if both $\mathcal{F}$ and $\mathcal{G}$ are non-trivial. (iii) $\min \{\mathrm{\mid }\mathcal{F}\setminus \mathcal{C}\mathrm{\mid },\mathrm{\mid }\mathcal{G}\setminus \mathcal{C}\mathrm{\mid }\}\le t((n-t-1)!-(n-t-2)!)$ for some $t$-coset $\mathcal{C}$ if $t\ge 2$. We also characterize all extremal configurations. The first result extends a theorem of Ellis (2011) to an exponentially wider range and sharpens the stability theorem of Keller, Lifshitz, Minzer and Sheinfeld (2024); the second gives a product version of the classical Hilton–Milner–Frankl theorem for permutations; and the third settles the remaining cases $t\ge 2$ of a conjecture of Ellis (2011) in a stronger form. In all three results, the linear dependence on $t$ is essentially optimal. Our proofs are based on the spread approximation method introduced by Kupavskii and Zakharov and on an approach to cross $t$-intersection problems developed by the present authors, with several essential refinements. As an application of our approach, we prove a product version of the Hilton–Milner–Frankl theorem for the alternating group.