On the $C$-diversity of intersecting hypergraphs

Let $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{X}{k}\right)$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$ for all $F,F^{\mathrm{\prime }}\in \mathcal{F}$. Let $\mathrm{\Delta }(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\ge 1$ the $C$-diversity, ${\gamma }_C(\mathcal{F})$ is defined as $\mathrm{\mid }\mathcal{F}\mathrm{\mid }-C\mathrm{\Delta }(\mathcal{F})$. Define ${\mathcal{F}}_{123}=\{F\in \left(\genfrac{}{}{0px}{}{X}{k}\right)\text{\hspace{0.17em}⁣}:\mathrm{\mid }F\cap \{1,2,3\}\mathrm{\mid }=2\}$. It has $C$-diversity $(3-2C)\left(\genfrac{}{}{0px}{}{n-3}{k-2}\right)$. The main result shows that for $1<C<\frac{3}{2}$ and $n\ge \frac{42}{3-2C}k$, ${\gamma }_C(\mathcal{F})\le {\gamma }_C({\mathcal{F}}_{123})$ 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.