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We describe a method that we believe may be foundational for a comprehensivetheory of generalised Turan problems. The cornerstone of our approach is aquasirandom counting lemma for quasirandom hypergraphs, which extends thestandard counting lemma by not only counting copies of a particularconfiguration but also showing that these copies are evenly distributed. Wedemonstrate the power of the method by proving a conjecture of Mubayi on thecodegree threshold of the Fano plane, that any 3-graph on n vertices for whichevery pair of vertices is contained in more than n/2 edges must contain a Fanoplane, for n sufficiently large. For projective planes over fields of odd sizeq we show that the codegree threshold is between n/2-q+1 and n/2, but forPG_2(4) we find the somewhat surprising phenomenon that the threshold is lessthan (1/2-c)n for some small c>0. We conclude by setting out a program forfuture developments of this method to tackle other problems.

A hypergraph regularity method for generalised Turan problems