Uniform hypergraphs with many edge‐colorings avoiding a fixed rainbow expanded complete graph

For fixed positive integers r and k , and a fixed k ‐uniform hypergraph F , we look for n ‐vertex hypergraphs that admit the maximum number of r ‐edge‐colorings with the property that there is no copy of F for which all hyperedges are assigned different colors. We consider F = H ℓ + 1 ( k ) , the k ‐uniform expanded complete graph, and we prove that the Turán hypergraph T ℓ ( k )( n ) , the balanced ℓ ‐partite k ‐uniform hypergraph on n vertices, is the only extremal configuration for any r ≥ r 0 ( k , ℓ ) and large n .