  In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can
be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is,
it can be colored with $O(\log n)$ colors such that for any (covered) point $p$
there is some disc whose color is distinct from all other colors of discs
containing $p$. They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
  \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be
colored with a total of at most $O(k \log n)$ colors such that (a) for any
point $p$ that is covered by at least $k$ discs, there are at least $k$
distinct discs each of which is colored by a color distinct from all other
discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs,
all discs covering $p$ are colored distinctively. We call such a coloring a
{\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
  \item [(ii)] More generally, for families of $n$ simple closed Jordan regions
with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists
a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors.
  \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be
$k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors.
  \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong
Conflict-Free coloring and the recently studied notion of $k$-colorful
coloring. \end{enumerate}
  All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings.
