Extremal bi-Helly families
\textit{V. I. Voloshin} [Australas. J. Comb. 11, 25-45 (1995; Zbl 0827.05027)] introduced the concept of \((p,q,s)\)-Helly hypergraph. A hypergraph is \(p\)-wise \(q\)-intersecting if the intersection of every \(p\) or fewer edges contains at least \(q\) vertices. A hypergraph \(\mathcal{H}\) is said to be \((p,q,s)\)-Helly (\(p\geq 2\) and \(q,s \geq 1\)) if \(|\bigcap_{H\in {\mathcal{H}}^\prime} H|\geq s\) holds for every \(p\)-wise \(q\)-intersecting subhypergraph \({\mathcal{H}}^\prime \subseteq \mathcal{H}\). The case \(p=2\) and \(q=s\) is studied in the paper. The main result states that the maximal number of \(r\)-tuples (edges) in a \(r\)-uniform \((2, k, k)\)-Helly family on \(n\) vertices is equal to \(\binom{n-k}{r-k}\) for all \(n\geq r >2k \geq 4\). For \(r=3,4\) some interesting asymptotic bounds are presented.
scientific article; zbMATH DE number 1420395