Extremal problems for colorings of simple hypergraphs and applications

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple hypergraph H with maximum edge degree at most Δ(H)≤c⋅nrn−1, is r-colorable, where c>0 is an absolute constant. As an application of our proof technique we establish a new lower bound for the Van der Waerden number W(n,r).