On 3-Hypergraphs with Forbidden 4-Vertex Configurations

Every 3-graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥4/9(1-o(1)). This bound is tight. The proof is a rather elaborate application of Cauchy–Schwarz-type arguments presented in the framework of flag algebras. We include further demonstrations of this method by re-proving a few known tight results about hypergraph Turán densities and significantly improving numerical bounds for several problems for which the exact value is not known yet.