The maximum $t$-diversity of $t$-intersecting families of permutations

Source abstract

The study of $t$-intersecting families in symmetric groups received lots of attention in the last two decades. In this paper, we study two different kinds of stability results for $t$-intersecting families in symmetric groups. Let $Σ_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. The $t$-diversity $γ_t(\mathcal{F})$ of $\mathcal{F}\subseteqΣ_n$ is defined as the minimum number of members of $\mathcal{F}$ whose deletion results in a family with transversal number $t$. The star $t$-diversity $γ_t^{\star}(\mathcal{F})$ is defined as the minimum number of members of $\mathcal{F}$ whose deletion results in a $t$-star. For $n$ relatively large with respect to $t$, we determine the best possible bounds for both $γ_t(\mathcal{F})$ and $γ_t^{\star}(\mathcal{F})$ over all $t$-intersecting families $\mathcal{F}\subseteqΣ_n$. The equality holding conditions are characterized. For $γ_t(\mathcal{F})$, the extremal case is generated by all $2t$-subsets of a $3t$-partial permutation or, when $t=2$, the family complements of the lines of the Fano plane. For $γ_t^{\star}(\mathcal{F})$, the extremal family is generated by all $(t+1)$-subsets of a $(t+2)$-partial permutation.