On -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 ) denote the minimum number of edges in an n -uniform non- r -colorable hypergraph. We prove that for all n ≥ 3 and r ≥ 3 , the following inequality holds m ( n , r ) ≥ 1 2 n 1 / 2 r n − 1 . This bound improves all the previously known bounds for r = 3 and ln ln n ≪ r ≪ ln n ln ln n . We also obtain some results concerning colorings of simple and l -simple hypergraphs. These results are based on our new lower bound for the maximum edge degree in an n -uniform non- r -colorable hypergraph.