A Non-aligning Variant of Generalized Turán Problems

In the so-called generalized Turán problems we study the largest number of copies of H in an n-vertex F-free graph G. Here we introduce a variant, where F is not forbidden, but we restrict how copies of H and F can be placed in G. More precisely, given an integer n and graphs H and F, what is the largest number of copies of H in an n-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of F? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the generalized Turán number for some pairs of graphs.