Rainbow Tur\'an problems for paths and forests of stars

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a {\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $e{x}^{\ast }(n,F)$, is the {\emph rainbow Tur'an number} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra"ete in 2007. We determine $e{x}^{\ast }(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $e{x}^{\ast }(n,F)$ when $F$ is a path with $k$ edges, disproving a conjecture in Keevash et al.