An extremal problem in proper (r, p)-coloring of hypergraphs

Let G(V,E) be a k-uniform hypergraph. A hyperedge e ∈ 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 rcoloring. An r-coloring of vertices in V is called a strong (r, p) coloring if every hyperedge e∈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.