Characterize maximum-sized non-centred families

Characterize, for every fixed alpha in (0,1) and sufficiently large n, every maximum-sized non-centred intersecting family of permutations in Sym(n,k) for k<=n^alpha, by proving that it is a Hilton--Milner family of the form {sigma in Sym(n,k): sigma(r)=s and sigma cap pi is nonempty} union {pi} for suitable r,s in [n] and pi in Sym(n,k) with pi(r) != s.

Background

The paper establishes upper bounds on non-centred intersecting families but does not determine their exact extremal structure. It introduces Hilton--Milner families by fixing an ordered pair (r,s), requiring agreement with a designated permutation pi at some point, and adjoining pi itself.

The authors state that an analogue of the exact characterization known for intersecting families in Sym(n) should hold for Sym(n,k) when k <= nalpha. Resolving this conjecture would strengthen the paper's stability results from asymptotic size estimates to a complete structural classification of extremal non-centred families.

References

We conjecture that these families are exactly the Hilton--Milner families that we constructed in \S~\ref{sec:der}. Given $\alpha \in (0,1)$, for all sufficiently large $n$ and $k\leq n{\alpha}$, if ${G} \subset Sym(n,k)$ is a non-centred intersecting family of the maximum possible size, then there exists $r,s\in [n]$ and $\pi \in Sym(n,k)$ with $\pi(r)\neq s$ such that {G}={\sigma \in Sym(n,k)\colon \ \sigma(r)=s\ \text{and}\ \sigma \cap \pi\neq \emptyset} \cup {\pi}.

— Intersecting families of permutations with a fixed number of cycles  (2608.20248 - Pantangi, 20 Aug 2026) in Section 6, Conclusion (Conjecture environment following the discussion of Ellis's result)