For any $\epsilon>0$ and $n>(1+\epsilon)t$, $n>n_0(\epsilon)$ we determine the size of the largest $t$-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) for $n>(1+\epsilon )t$. We note that, for this range of parameters, the extremal examples are not necessarily trivial, and that our statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The proof is based on the refinement of the method of spread approximations, recently introduced by Kupavskii and Zakharov (2022).

翻译：对于任意$\epsilon>0$及$n>(1+\epsilon)t$，$n>n_0(\epsilon)$，我们确定了最大$t$-交排列族的大小，并给出了一个锐利的稳定性结果。这解决了Ellis、Friedgut和Pilpel（2011）的一个猜想，并验证了Frankl和Deza（1977）以及Cameron（1988）的猜想在$n>(1+\epsilon)t$条件下的有效性。我们注意到，在此参数范围内，极值示例并非一定是平凡的，并且我们的结论类似于著名的Ahlswede-Khachatrian定理。该证明基于对Kupavskii和Zakharov（2022）近期提出的扩散近似方法的改进。