Complete Partite subgraphs in dense hypergraphs


  • Supported by NSF DMS 0800070; Heisenberg-Programme of the Deutsche Forschungsgemeinschaft (DFG Grant SCHA 1263/4-1); Emory University Research grant.

    Part of this work was carried out at the Institute of Advanced Studies in Princeton during the special semester on Arithmetic Combinatorics in Fall 2007 organized by J. Bourgain and V. Vu.


For a given r -uniform hypergraph equation image we study the largest blow-up of equation image which can be guaranteed in every large r -uniform hypergraph with many copies of equation image. 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(log n) 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.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012