  We consider extremal edge-coloring problems inspired by the theory of
anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free
colorings.
  Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n
\geq |G|$, let $f(n,G|F)$ denote the smallest integer $k$ such that any edge
coloring of the complete graph $K_n$ with at least $k$ colors forces a copy of
$G$ in which each color class induces a member of $F$. Observe that in
anti-Ramsey problems each color class is a single edge; i.e., $F=\{K_2\}$.
  In our previous paper [arXiv:2405.19812], attention was given mostly to the
case where $F$ is hereditary under subgraph inclusion. In the present work we
consider coloring problems inspired by odd-coloring and conflict-free coloring.
As we shall see, dealing with these problems requires distinct additional tools
to those used in our first paper on the subject.
  Among the many results introduced in this paper, we mention:
  (1) For every graph $G$, there exists a constant $c=c(G)$ such that in any
edge coloring of $K_n$ with at least $cn$ colors there is a copy of $G$ in
which every vertex $v$ is incident with an edge whose color appears only once
among all edges incident with $v$.
  (2) In sharp contrast to the above result we prove that if $F$ is the class
of all odd graphs (having vertices with odd degrees only) then
$f(n,K_k|F)=(1+o(1))$ex$(n,K_{\lceil k/2 \rceil})$, which is quadratic for $k
\geq 5$.
  (3) We exactly determine $f(n,G|F)$ for small graphs when $F$ belongs to
several families representing various odd/even coloring constraints.
