Colorings of b-simple hypergraphs

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of b-simple hypergraphs, in which any two edges do not share more than b common vertices. We prove that for n≥n0(b), any n-uniform b-simple hypergraph with the maximum edge degree at most cnrn−b is r-colorable, where c>0 is an absolute constant.