$t$-相交置换族的极大 $t$-多样性

中文摘要 AI 辅助

在过去二十年中，对称群中 $t$-相交族的研究受到了广泛关注。本文研究了对称群中 $t$-相交族的两种不同类型的稳定性结果。设 $\Sigma_n$ 表示集合 $\{1,2,\ldots,n\}$ 上的对称群。$\mathcal{F}\subseteq\Sigma_n$ 的 $t$-多样性 $\gamma_t(\mathcal{F})$ 定义为删除 $\mathcal{F}$ 中某些成员后得到一个横截数为 $t$ 的族所需删除的最少成员数。星形 $t$-多样性 $\gamma_t^{\star}(\mathcal{F})$ 定义为删除 $\mathcal{F}$ 中某些成员后得到一个 $t$-星形所需删除的最少成员数。对于相对于 $t$ 足够大的 $n$，我们确定了所有 $t$-相交族 $\mathcal{F}\subseteq\Sigma_n$ 上 $\gamma_t(\mathcal{F})$ 和 $\gamma_t^{\star}(\mathcal{F})$ 的最佳可能界。刻画了等号成立的条件。对于 $\gamma_t(\mathcal{F})$，极值情形由所有 $2t$-子集生成的 $3t$-部分置换产生，或者当 $t=2$ 时，由 Fano 平面线的族的补集产生。对于 $\gamma_t^{\star}(\mathcal{F})$，极值族由所有 $(t+1)$-子集生成的 $(t+2)$-部分置换产生。

英文摘要

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.

发表机构

• School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
• School of Mathematics, Sichuan University（四川大学数学学院）
• School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）

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