On Hypergraphs with Every Four Points Spanning at Most Two Triples - Mubayi, Dhruv (The Electronic Journal of Combinatorics, 2003)
Let ${\cal F}$ be a triple system on an $n$ element set. Suppose that ${\cal F}$ contains more than $(1/3-ε){n\choose 3}$ 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 ${\cal F}$. This improves previous bounds of de Caen and Matthias.