On rr-chromatic hypergraphs

This work deals with a classical combinatorial problem of P. Erdős and A. Hajnal concerning colorings of uniform hypergraphs. Let m(n,r)m(n,r) denote the minimum number of edges in an nn-uniform non-rr-colorable hypergraph. We prove that for all n≥3n≥3 and r≥3r≥3, the following inequality holds m(n,r)≥12n1/2rn−1. This bound improves all the previously known bounds for r=3r=3 and lnlnn≪r≪lnnlnlnn. We also obtain some results concerning colorings of simple and ll-simple hypergraphs. These results are based on our new lower bound for the maximum edge degree in an nn-uniform non-rr-colorable hypergraph.