AI 中文总结
本文研究固定循环数的置换集合Sym(n,k)中最大相交族的结构，证明k≤n^0.25时最大相交族为星族，还给出非中心相交族大小的稳定性界并在多对数范围得到渐近紧的结果，建立了对应的Erdős–Ko–Rado定理及稳定性版本。
AI 中文摘要
设$\text{Sym}(n,k)$表示集合$\text{{1,2,…,n}}$上恰好有$k$个循环的置换构成的集合。若族$\text{Sym}(n,k)$满足对所有$σ,τ∈\text{Sym}(n,k)$，$σ^{-1}τ$都有一个不动点，则称其为相交族。本文研究$\text{Sym}(n,k)$中最大规模相交族的大小与结构。在$k≤n^{0.25}$的范围内，我们证明每个最大规模相交族都是星族，即由$\text{Sym}(n,k)$中在$[n]$的某个给定点上取值一致的所有置换构成。我们通过证明一个更强的稳定性结果得到该结论，该结果给出了非中心相交族的最大可能大小的界。具体而言，在$k≤n^{0.25}$的范围内，任何非中心相交族的大小至多为星族最大可能大小的$(2/3+o(1))$倍。在更严格的多对数范围$k≤(\text{ln}\text{ }n)^d$内，我们将该界改进为星族最大可能大小的$(1-1/e+o(1))$倍；我们证明该界是渐近紧的。由此，我们建立了$\text{Sym}(n,k)$的Erdős–Ko–Rado定理及其对应的稳定性版本。
英文摘要
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $σ^{-1}τ$ has a fixed point for all $σ,τ\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{Sym}(n,k)$. In the regime $k\leq n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{Sym}(n,k)$ that agree at a given point in $[n]$. We establish this result by proving a stronger stability result that bounds the maximum possible size of a non-centred intersecting family. Specifically, in the regime $k\leq n^{0.25}$, the size of any non-centred intersecting family is at most $\left(2/3+o(1)\right)$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\leq (\ln n)^{d}$, we improve this bound to $\left(1-1/e+o(1)\right)$ times the maximum possible size of a star; we show that this bound is asymptotically sharp. Thus, we establish both an Erdős--Ko--Rado theorem and its corresponding stability version for $\mathrm{Sym}(n,k)$.
