| Abstract | A classic result of Rademacher from 1941 led to the study of supersaturation problems of graphs, which aim to count the minimum number of copies of a given graph F among all graphs with n vertices and m edges. This is closely related to a central concept in Extremal Graph Theory -- the Turán number of F. Famous results of Erdős, and Lovász and Simonovits determine the minimum number of cliques Kr in graphs whose number of edges exceed the Turán number of Kr by a linear term O(n). Subsequent works of Mubayi as well as Pikhurko and Yilma extend these classical results from cliques to color-critical graphs, a rich family playing important roles in extremal problems. In this talk, we will discuss supersaturation problems beyond color-critical graphs and investigate natural enumerative parameters. Our results go beyond the previous results and show that supersaturation problems for general graphs can be rather complicate. Among others, we disprove a conjecture of Mubayi. Joint work with Long-Tu Yuan. |
