A note on the maximum diversity of intersecting families in the symmetric group

Abstract

Let ${\mathcal{S}}_n$ be the symmetric group on the set $[n]:=\{1,2,\dots ,n\}$. A family $\mathcal{F}\subset {\mathcal{S}}_n$ is called intersecting if for every $\sigma ,\pi \in \mathcal{F}$ there exists some $i\in [n]$ such that $\sigma (i)=\pi (i)$. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of permutations in $\mathcal{F}$, whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for $n\ge 500$ the diversity of an intersecting subfamily of ${\mathcal{S}}_n$ is at most $(n-3)(n-3)!$, which is best possible.

Final version