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.
Categories: math.CO