Generalized Turan Problems for Small Graphs

For graphs H and F, the generalized Turan number ex(n, H, F) is the largest number of copies of H in an F -free graph on n vertices. We consider this problem when both H and F have at most four vertices. We give sharp results in almost all cases, and connect the remaining cases to well-known unsolved problems. Our main new contribution is applying the progressive induction method of Simonovits for generalized Turan problems.