Improved algorithms 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 (i.e. every two distinct edges share at most one vertex) hypergraph $H$ with maximum edge degree at most $Δ(H)\leq c\cdot nr^{n-1},$ is $r$-colorable, where $c>0$ is an absolute constant. %We prove also that similar result holds for $b$-simple hypergraphs. As an application of our proof technique we establish a new lower bound for Van der Waerden number $W(n,r)$, the minimum $N$ such that in any $r$-coloring of the set $\{1,...,N\}$ there exists a monochromatic arithmetic progression of length $n$. We show that $W(n,r)>c\cdot r^{n-1},$ for some absolute constant $c>0$.

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