Supersaturation problem for color-critical graphs

The Turán function ex(n, F ) of a graph F is the maximum number of edges in an F -free graph with n vertices. The classical results of Turán and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine hF (n, q), the minimum number of copies of F that a graph with n vertices and ex(n, F ) + q edges can have. We determine hF (n, q) asymptotically when F is color-critical (that is, F contains an edge whose deletion reduces its chromatic number) and q = o(n). Determining the exact value of hF (n, q) seems rather difficult. For example, let c1 be the limit superior of q/n for which the extremal structures are obtained by adding some q edges to a maximal F -free graph. The problem of determining c1 for cliques was a well-known question of Erdős that was solved only decades later by Lovász and Simonovits. Here we prove that c1 > 0 for every color-critical F . Our approach also allows us to determine c1 for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.