Intersecting families of permutations with a fixed number of cycles

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 $\mathrm{Sym}(n,k)$.