A note on the maximum diversity of intersecting families in the symmetric group
Jian Wang, Jimeng Xiao
TL;DR
This work determines the maximum diversity of intersecting families of permutations in the symmetric group $\mathcal{S}_n$. By applying the spread approximation framework of Kupavskii and Zakharov, the authors show that for $n \ge 500$, any intersecting subfamily $\mathcal{F} \subset \mathcal{S}_n$ satisfies $\gamma(\mathcal{F}) \le (n-3)(n-3)!$, and this bound is tight, achieved by the triangle family $\mathcal{T}(n)$. The approach uses $r$-spreadness, the sets $\mathcal{F}(A)$, and the pseudo-sunflower technique (Füredi), combined with a structured basis construction to decompose $\mathcal{F}$ and control the diversity. These results extend Erdos–Ko–Rado-type phenomena to permutation groups and identify the extremal configuration for diversity in this setting.
Abstract
Let $\mathcal{S}_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. A family $\mathcal{F}\subset \mathcal{S}_n$ is called intersecting if for every $σ,π\in \mathcal{F}$ there exists some $i\in [n]$ such that $σ(i)=π(i)$. 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 $n\geq 500$ the diversity of an intersecting subfamily of $\mathcal{S}_n$ is at most $(n-3)(n-3)!$, which is best possible.