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

TL;DR

This paper investigates the maximum diversity of intersecting families within the symmetric group, establishing an upper bound for large n using advanced combinatorial methods, and confirming the bound's optimality.

Contribution

It introduces a new upper bound on the diversity of intersecting families in symmetric groups for large n, utilizing the spread approximation method.

Findings

Maximum diversity bound of (n-3)(n-3)! for n ≥ 500

The bound is proven to be tight and optimal

Application of the spread approximation method to permutation groups

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 $\sigma,\pi\in \mathcal{F}$ 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 $n\geq 500$ the diversity of an intersecting subfamily of $\mathcal{S}_n$ is at most $(n-3)(n-3)!$, which is best possible.