Extremal graphs in the generalized Turán problem

One of the first results in extremal graph theory is Turán's theorem, which states that the Turán graph T(n,r) maximizes the number of edges in an n-vertex graph that does not contain a clique of size r+1 as a subgraph. More generally, given two graphs H and F, the generalized Turán number ex(n, H, F) is the largest number of copies of H in an n-vertex F-free graph and such graphs with ex(n, H, F) copies of H are called extremal graphs. For fixed H, F=K_{r+1} and r large enough, we showed in a joint work with Morrison, Nir, Norine, and Rzążewski, that the extremal graph for the generalized Turán problem is the Turán graph T(n,r). This talk discusses this result and more broadly, results on generalized Turán numbers.