Transversals in uniform hypergraphs with property (7, 2) (Q1817581)

Let \(f(r,p,2)\) \((p> 1, r\geq 2)\) be the maximum of the cardinality of a minimum transversal over all \(r\)-uniform hypergraphs \(\mathcal{H}\) possessing the property that every subhypergraph of \(\mathcal{H}\) with \(p\) edges has a transversal of size \(2\). The values of \(f(r,p,2)\) for \(p=3,4,5,6\) were found by \textit{P. Erdős, D. Fon-Der-Flaass, A. V. Kostochka} and \textit{Zs. Tuza} [Sib. Adv. Math. 2, No. 1, 82-88 (1992; Zbl 0848.05049)]. The case \(p=7\) is much harder. The authors give some interesting bounds on \(f(r,7,2)\). In particular, they show that \(f(4k,7,2)\geq 3k+1\) for \(k \geq 10\) and that \(f(r,7,2)\leq \lceil 7r/8\rceil\).