An almost complete $t$-intersection theorem for permutations

TL;DR

This paper determines the maximum size of $t$-intersecting permutation families for large $n$, confirming longstanding conjectures and extending the Ahlswede-Khachatrian theorem to permutations with stability results.

Contribution

It provides a near-complete $t$-intersection theorem for permutations for large $n$, resolving conjectures and introducing refined spread approximation methods.

Findings

Exact size of largest $t$-intersecting permutation families for $n>(1+psilon)t$

Sharp stability results for these families

Validation of conjectures by Ellis, Friedgut, Pilpel, Frankl, Deza, and Cameron

Abstract

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).