<正> The lower bounds for any R(l1,…,lq;r)are investigated.Let K_nr be the completer-uniform hypergraph on n points.Define R(l1,…,lq;r)as the minimal natural number n sothat if the edges of K_nr are q-colored,there is a set S of li(i∈{1,…,q})vertices such that alledges on S are of the i-th color.For the special case of q=r=2,the lower bounds were got byP.Erd(?)s and J.Spencer.In this paper,we shall give the lower bounds for any R(l1,…,lq;r).