A hypergraph regularity method for generalized Turán problems

Wedescribe amethod that we believemay be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed.We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2− )n for some small > 0. We conclude by setting out a program for future developments of this method to tackle other problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2009