  Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be
properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains
vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring
of vertices in $V$ is called a {\it strong $(r,p)$ coloring} if every hyperedge
$e \in E$ is properly $(r,p)$ colored by the $r$-coloring. We study the maximum
number of hyperedges that can be properly $(r,p)$ colored by a single
$r$-coloring and the structures that maximizes number of properly $(r,p)$
colored hyperedges.
