The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let denote the minimum possible number of edges in an -uniform non--colorable hypergraph. We show that for , where are some absolute constants. Moreover, we obtain similar bounds for , which is equal to the minimum possible value of the maximum edge degree in an -uniform non--colorable hypergraph. If , then where are some other absolute constants.