Some exact results for generalized Turán problems

Fix a k-chromatic graph F. In this paper we consider the question to determine for which graphs H does the Turán graph T_k−1(n) have the maximum number of copies of H among all n-vertex F-free graphs (for n large enough). We say that such a graph H is F-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph H, there is k large enough such that H is K_k-Turán-good. (ii) The path P₃ is F-Turán-good for F with χ(F)≥4. (iii) The path P₄ and cycle C₄ are C₅-Turán-good. (iv) The cycle C₄ is F₂-Turán-good where F₂ is the graph of two triangles sharing exactly one vertex.