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The paper deals with extremal problems concerning colorings of hypergraphs.By using a random recoloring algorithm we show that any $n$-uniform simple(i.e. every two distinct edges share at most one vertex) hypergraph $H$ withmaximum edge degree at most \[  \Delta(H)\leq c\cdot nr^{n-1}, \] is $r$-colorable, where $c>0$ is anabsolute constant. %We prove also that similar result holds for $b$-simplehypergraphs.  As an application of our proof technique we establish a new lower bound forVan der Waerden number $W(n,r)$, the minimum $N$ such that in any $r$-coloringof the set $\{1,...,N\}$ there exists a monochromatic arithmetic progression oflength $n$. We show that \[  W(n,r)>c\cdot r^{n-1}, \] for some absolute constant $c>0$.

Improved algorithms for colorings of simple hypergraphs and applications