On the C-diversity of intersecting k-graphs

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