For k-uniform hypergraphs F and H and an integer , let denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let , where the maximum is taken over the family of all k-uniform hypergraphs on n vertices. Moreover, let be the usual extremal function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F. Here, we consider the question for determining for F being the k-uniform expanded, complete graph or the k-uniform Fan(k)-hypergraph with core of size , where , and we show
for and n large enough. Moreover, for or , for k-uniform hypergraphs H on n vertices, the equality only holds if H is isomorphic to the ℓ-partite, k-uniform Turán hypergraph on n vertices, once n is large enough. On the other hand, we show that is exponentially larger than , if .