A Localized Approach to Generalized Turán Problems

Generalized Turán 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 Bradač, Malec, and Tompkins to generalized Turán problems. We weight the copies of H (typically taking H=Kₜ), 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 Turán 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, K₁, ₑ) for every H having at least one dominating vertex and mex (m, H, K₁, ₑ) for every H having at least two dominating vertices.