On the $C$-diversity of intersecting hypergraphs

Abstract

Let $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{X}{k}\right)$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{F}$. Let $\hat{I}”(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\ge 1$ the $C$-diversity, ${\hat{I}}_C^3(\mathcal{F})$ is defined as $|\mathcal{F}|-C\hat{I}”(\mathcal{F})$. Define ${\mathcal{F}}_{123}=\left\{F\in \left(\genfrac{}{}{0ex}{}{X}{k}\right):|F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\left(\genfrac{}{}{0ex}{}{n-3}{k-2}\right)$. The main result shows that for $1<C<\frac{3}{2}$ and $n\ge \frac{42}{3-2C}k$, ${\hat{I}}_C^3(\mathcal{F})\le {\hat{I}}_C^3({\mathcal{F}}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to ${\mathcal{F}}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven.

17 pages