Best possible bounds on the double-diversity of intersecting hypergraphs
Peter Frankl, Jian Wang
TL;DR
The paper investigates tight bounds on the double-diversity \(\gamma_2(\mathcal{F})\) for intersecting \(k\)-uniform hypergraphs and related higher-diversity analogues. It introduces the Fano-based construction \(\mathcal{F}_{\mathcal{L}}\) and proves that, for \(n \ge 13k^2\), this family (up to isomorphism) uniquely maximizes \(\gamma_2\) among all intersecting \(k\)-graphs with no disjoint edges, with a precise extremal formula in terms of binomial coefficients. The work further develops exact results for triple-diversity, establishing \(m_3(4)=3\) and presenting several extremal 4-uniform examples achieving \(\gamma_3 = 3\). Beyond these specific bounds, the authors derive general inequalities relating gaps, transversals, and diversity (via a branching-process framework and wreath-product constructions) to map the landscape of extremal diversity problems and propose sharp conjectures and open questions, including potential improvements of constants and asymptotic behavior of a hierarchy of diversity parameters.
Abstract
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 $γ_2(\mathcal{F})$ 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.