For a given r -uniform hypergraph \$\{\textbackslash{}cal F\}\$ we study the largest blow-up of \$\{\textbackslash{}cal F\}\$ which can be guaranteed in every large r -uniform hypergraph with many copies of \$\{\textbackslash{}cal F\}\$. For graphs this problem was addressed by Nikiforov, who proved that every n -vertex graph that contains O(nl) copies of the complete graph Kl must contain a complete l -partite graph with O(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 Erdos and Hajnal.(c) 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012