A generalization of a Ramsey-type theorem on hypermatchings

For an r-uniform hypergraph G define N(G, l; 2) (N(G, l;Zn)) as the smallest integer for which there exists an r-uniform hypergraph H on N(G, l; 2) (N(G, l;Zn)) vertices with clique(H)< l such that every 2-coloring (Zn-coloring) of the edges of H implies a monochromatic (zero-sum) copy of G. Our results strengthen a Ramsey-type theorem of Bialostocki and Dierker on zero-sum hypermatchings. As a consequence we show that for any n ≥ 2, r ≥ 2, and l > r + 1, N(nKr r , l; 2) = N(nKr r , l;Zn) = (r + 1)n− 1.