Generalized rainbow Tur\'an problems

Alon and Shikhelman initiated the systematic study of the following generalized Tur'an problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Tur'an number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Tur'an problems was initiated by Keevash, Mubayi, Sudakov and Verstra"ete. Motivated by the above problems, we study the following problem: What is the maximum number of copies of $F$ in a properly edge-colored graph on $n$ vertices without a rainbow copy of $F$? We establish several results, including when $F$ is a path, cycle or tree.