Generalized rainbow Turán problems (Q2144330)

Summary: \textit{N. Alon} and \textit{C. Shikhelman} [J. Comb. Theory, Ser. B 121, 146--172 (2016; Zbl 1348.05100)] 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 \textit{P. Keevash} et al. [Comb. Probab. Comput. 16, No. 1, 109--126 (2007; Zbl 1119.05058)]. 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.