  Let $f(K_n, H, q)$ be the minimum number of colors needed to edge-color $K_n$
so that every copy of $H$ is colored with at least $q$ colors. Originally posed
by Erd\H{o}s and Shelah when $H$ is complete, the asymptotics of this extremal
function have been extensively studied when $H$ is a complete graph or a
complete balanced bipartite graph. Here we investigate this function for some
other $H$, and in particular we determine the asymptotic behavior of $f(K_n,
P_v, q)$ for almost all values of $v$ and $q$, where $P_v$ is a path on $v$
vertices.
