A hypergraph regularity method for generalized Turán problems

We describe a method that we believe may 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 PG 2 (4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − ϵ) n for some small ϵ > 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., 2009