Edge-colorings of uniform hypergraphs avoiding monochromatic matchings

For positive integers kk, ℓℓ and rr, and an rr-uniform hypergraph HH, let ck,ℓ,r(H)ck,ℓ,r(H) be the number of kk-colorings of the set of hyperedges of HH with no ℓℓ independent hyperedges of the same color. Let Hn,rHn,r denote the set of all nn-vertex rr-uniform hypergraphs and consider the function ck,ℓ,r(n)=max  {ck,ℓ,r(H):H∈Hn,r}ck,ℓ,r(n)=max  {ck,ℓ,r(H):H∈Hn,r}, the maximum of ck,ℓ,r(H)ck,ℓ,r(H) over all rr-uniform hypergraphs HH on nn vertices. In this paper, we determine, for all fixed values of rr, kk and ℓℓ, and large nn, the rr-uniform nn-vertex hypergraphs HH with ck,ℓ,r(n)=ck,ℓ,r(H)ck,ℓ,r(n)=ck,ℓ,r(H).