On the rainbow planar Turán number of paths

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by ${\text{ex}}_{\text{p}}^{\ast }(n,F)$. In particular, we determine ${\text{ex}}_{\text{p}}^{\ast }(n,P_5)$, where $P_5$ denotes the $5$-vertex path.