On hypergraphs with every four points spanning at most two triples
Summary: Let \(\mathcal F\) be a triple system on an \(n\)-element set. Suppose that \(\mathcal F\) contains more than \((1/3-\varepsilon)\left ( \begin{smallmatrix} n \\ 3 \end{smallmatrix}\right) \) triples, where \(\varepsilon > 10^{-6}\) is explicitly defined and \(n\) is sufficiently large. Then there is a set of four points containing at least three triples of \(\mathcal F\). This improves previous bounds of \textit{D. de Caen} [Ars Comb. 16, 5-10 (1983; Zbl 0532.05037)] and \textit{U. Matthias} [Hypergraphs without complete \(r\)-partite subgraphs (German) (Heidelberg: Univ., Naturwiss.-Mathemat. Gesamtfak.) (1994; Zbl 0818.05048)].
scientific article; zbMATH DE number 1985385