A localized approach to generalized Turan problems

Generalized Turan problems ask for the maximum number of copies of a graph H in an n-vertex, F-free graph, denoted by ex(n, H, F ). We show how to extend the new, localized approach of Bradac, Malec, and Tompkins to generalized Turan problems. We weight the copies of H (typically taking H = K- t ), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of H , and in each case prove a tight upper bound on the sum of the weights. The generalized edge Turan number mex(m, H, F ) is the maximum number of copies of a graph H in an m-edge, F-free graph. A consequence of our new localized theorems is an asymptotic determination of ex(n, H, K (1,r) ) for every H having at least one dominating vertex and mex(m, H, K (1,r) ) for every H having at least two dominating vertices.