Abstract: Bondy et al. showed that any tripartite graph in which the density of edges between each pair of vertex classes is at least 0.618.. (the golden ratio) must contain a triangle, moreover this is sharp. The density Turán problem for a graph or hypergraph H asks for the corresponding quantity guaranteeing a copy of H in a subgraph of a blow-up of H. Previous results due to Csikvári and Nagy gave bounds for graphs related to the largest root of the matching polynomial; while the only exact result for hypergraphs is for the complete r-graph of order r+1, due to Markström and Thomassen.
In joint work with Adam Sanitt we use an entropy compression argument to provide upper bounds for any hypergraph.
