Applications of the regularity lemma for uniform hypergraphs

In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n) copies of K (k) t . Then one can delete o(n) edges of H to make it K (k) t -free. Similar results were recently obtained by W. T. Gowers.