On Hypergraphs with Every Four Points Spanning at Most Two Triples - Mubayi, Dhruv (The Electronic Journal of Combinatorics, 2003)
Let F be a triple system on an n element set. Suppose that F contains more than (1/3-ε)n3 triples, where ε>10^-6 is explicitly defined and n is sufficiently large. Then there is a set of four points containing at least three triples of F. This improves previous bounds of de Caen and Matthias.