Covering complete partite hypergraphs by monochromatic components

Extending our results achieved for ℓ=1, we prove that for any r≥3,2≤ℓ≤r,k≥1+r−ℓ, in every spanning k-coloring of the edges of a complete r-uniform (r,ℓ)-partite hypergraph, the vertex set can be covered by at most 1+⌊k−r+ℓ−1ℓ⌋ sets, each forming a connected hypergraph in some color.