Extend the sharp stability bound to polynomially growing cycle counts

Prove that for every fixed alpha in (0,1) and every xi>0, every sufficiently large non-centred intersecting family G of permutations on [n] with exactly k cycles, where k is at most n^alpha, satisfies |G| <= (1-1/e+xi) max{c(n-1,k),c(n-1,k-1)}.

Background

The paper studies intersecting families in Sym(n,k), the set of permutations of [n] having exactly k cycles, where two permutations intersect when they agree at some point. It proves an EKR theorem and stability bounds for k <= n0.25, obtaining a non-centred-family bound of approximately 2/3 of the largest star size, and improves this to approximately 1-1/e when k <= (ln n)d.

The construction of Hilton--Milner families shows that the factor 1-1/e is asymptotically sharp in the polylogarithmic regime. The authors explain that their Bonferroni-based estimates only establish the corresponding upper bound up to the exponent alpha=0.25, although other components of the method extend to arbitrary fixed alpha in (0,1). The conjecture asks for the sharp 1-1/e stability estimate throughout the polynomial regime k <= nalpha.

References

We conjecture that the sharp bound in Theorem~\ref{thm:stabilityln} can be extended to $k\leq n{\alpha}$ for $\alpha \in (0,1)$. Given $\xi>0$ and $\alpha \in (0,1)$, for all sufficiently large $n$ and $k\leq n{\alpha}$, if ${G} \subset Sym(n,k)$ is a non-centred intersecting family, then |{G}|\leq \left(1-1/e +\xi \right) \max{c(n-1,k),c(n-1,k-1)}.

— Intersecting families of permutations with a fixed number of cycles  (2608.20248 - Pantangi, 20 Aug 2026) in Section 6, Conclusion (following Theorem 2.2; Conjecture environment)