Best possible bounds on the double-diversity of intersecting hypergraphs

For a family $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\text{\hspace{0.17em}⁣}:x\mathrm{\notin }F,y\mathrm{\notin }F\}$. The double-diversity ${\gamma }_2(\mathcal{F})$ is defined as the minimum of $\mathrm{\mid }\mathcal{F}(\bar{x},\bar{y})\mathrm{\mid }$ over all pairs $x,y$. Let $\mathcal{L}\subset \left(\genfrac{}{}{0px}{}{[7]}{3}\right)$ consist of the seven lines of the Fano plane. For $n\ge 7$, $k\ge 3$ one defines the Fano $k$-graph ${\mathcal{F}}_{\mathcal{L}}$ as the collection of all $k$-subsets of $[n]$ that contain at least one line. It is proven that for $n\ge 13k^2$ the Fano $k$-graph is the essentially unique family maximizing the double diversity over all $k$-graphs without a pair of disjoint edges. Some similar, although less exact results are proven for triple and higher diversity as well.