On the extremal graphs in generalized Turán problems

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. For every graph F, we present an extremal graph for a generalized Turán problem. More precisely, we present a graph H not containing F and for sufficiently large n we present an F-free n-vertex graph G that contains exactly ex(n,H,F) copies of H.