Best possible bounds on the double-diversity of intersecting hypergraphs

For a family $\mathcal{F}\subset \binom{[n]}{k}$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\colon x\notin F,\ y\notin F\}$. The double-diversity $\gamma_2(\mathcal{F})$ is defined as the minimum of $|\mathcal{F}(\bar{x},\bar{y})|$ over all pairs $x,y$. Let $\mathcal{L}\subset\binom{[7]}{3}$ consist of the seven lines of the Fano plane. For $n\geq 7$, $k\geq 3$ 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 $n\geq 13k^2$ 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.