Weighted Turán densities for hypergraphs

Bondy and Tuza generalized the Turán problem for graphs to weighted graphs. Subsequently, Füredi and Kündgen gave a solution in the case of cliques. We consider the weighted Turán problem on hypergraphs and prove the first tight asymptotic result for 3-uniform hypergraphs. Specifically, we determine the weighted Turán density of the generalized triangle $F_5 = \{abc, abd, cde\}$. We also prove weighted stability results and exact results. Many intriguing open problems remain, the most tantalizing of which is to determine the weighted Turán density of the tetrahedron. Bounds for this density using flag algebras are presented.