Intersecting families of permutations with a fixed number of cycles

Let $\mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{n},\mathrm{k})$ denote the set of permutations on $\{1,2,\dots ,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset \mathrm{S}\mathrm{y}\mathrm{m}(n,k)$ is said to be intersecting if ${\sigma }^{-1}\tau$ has a fixed point for all $\sigma ,\tau \in \mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{S}\mathrm{y}\mathrm{m}(n,k)$. In the regime $k\le n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{S}\mathrm{y}\mathrm{m}(n,k)$ 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 $k\le n^{0.25}$, the size of any non-centred intersecting family is at most $(2\mathrm{/}3+o(1))$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\le (\ln n{)}^d$, we improve this bound to $(1-1\mathrm{/}e+o(1))$ 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 $\mathrm{S}\mathrm{y}\mathrm{m}(n,k)$.