# Generalized Rainbow Turán Numbers

### Abstract

Alon and Shikhelman [*J. Comb. Theory, B.* **121** (2016)] initiated the systematic study of the following generalized Turán 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 Turán 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 Turán problems was initiated by Keevash, Mubayi, Sudakov and Verstraete [*Comb. Probab. Comput. ***16** (2007)].

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.