Coloring hypergraphs with bounded cardinalities of edge intersections

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 $n\geqslant n_0(b)$, any $n$-uniform $b$-simple hypergraph with the maximum edge degree at most $c\cdot nr^{n-b}$ is $r$-colorable, where $c>0$ is an absolute constant. We also establish some applications of the main result.