Complete Partite subgraphs in dense hypergraphs

For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K` must contain a complete `-partite graph with Ω(logn) vertices in each class. We give another proof of Nikiforov’s result, make very small progress towards that problem for hypergraphs, and consider a Ramsey-type problem related to a conjecture of Erdős and Hajnal.