Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erdős (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n   is sufficiently large and 1⩽q