Title: A generalization of diversity for intersecting families Show Chinese title

Title: 相交族的多样性推广

Abstract: Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets and let $\Delta(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The \emph{diversity} of $\mathcal{F}$ is defined as $d(\mathcal{F}) := |\mathcal{F}| - \Delta(\mathcal{F})$. Diversity can be viewed as a measure of distance from the `trivial' maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed, the diversity of this family is $0$. Moreover, the diversity of the largest non-trivial intersecting family \`a la Hilton-Milner is $1$. It is known that the maximum possible diversity of an intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is $\binom{n-3}{r-2}$ as long as $n$ is large enough. We introduce a generalization called the \emph{$C$-weighted diversity} of $\mathcal{F}$ as $d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot \Delta(\mathcal{F})$. We determine the maximum value of $d_C(\mathcal{F})$ for intersecting families $\mathcal{F} \subseteq \binom{[n]}{r}$ and characterize the maximal families for $C\in \left[0,\frac{7}{3}\right)$ as well as give general bounds for all $C$. Our results imply, for large $n$, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl's Delta-system method. Show Chinese abstract

Abstract: 设$\mathcal{F}\subseteq \binom{[n]}{r}$为一个集合的相交族，令$\Delta(\mathcal{F})$为$\mathcal{F}$中的最大度数，即包含固定顶点的$\mathcal{F}$的边的最大数量。 \emph{多样性}是$\mathcal{F}$的定义为$d(\mathcal{F}) := |\mathcal{F}| - \Delta(\mathcal{F})$。 多样性可以看作是从由 Erdős-Ko-Rado 定理给出的“平凡”的最大大小相交族的距离的度量。 确实，这个族的多样性是$0$。 此外，Hilton-Milner 类型的最大非平凡相交族的多样性是$1$。 已知相交族的最大可能多样性$\mathcal{F}\subseteq \binom{[n]}{r}$在$n$足够大时是$\binom{n-3}{r-2}$。 我们引入了一个称为\emph{$C$加权多样性}的推广，即$\mathcal{F}$作为$d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot \Delta(\mathcal{F})$。 我们确定了对于相交族$\mathcal{F} \subseteq \binom{[n]}{r}$的$d_C(\mathcal{F})$的最大值，并对$C\in \left[0,\frac{7}{3}\right)$的极大族进行了表征，同时还给出了所有$C$的一般界。 我们的结果表明，对于大的$n$，一个关于相关多样性度量的Frankl和Wang的近期猜想成立。 我们主要的技术是一种Frankl的Delta系统方法的变体。