Colorings of hypergraphs with large number of colors

The paper deals with the well-known problem of Erdős and Hajnal concerning colorings of uniform hypergraphs and some related questions. Let m(n,r)m(n,r) denote the minimum possible number of edges in an nn-uniform non-rr-colorable hypergraph. We show that for r>nr>n, c1nlnn⩽m(n,r)rn⩽C1n3lnn, where c1,C1>0c1,C1>0 are some absolute constants. Moreover, we obtain similar bounds for d(n,r)d(n,r), which is equal to the minimum possible value of the maximum edge degree in an nn-uniform non-rr-colorable hypergraph. If r>nr>n, then c2nlnn⩽d(n,r)rn−1⩽C2n3lnn, where c2,C2>0c2,C2>0 are some other absolute constants.