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On proper colorings of hypergraphs

Let $\mathcal{H}$ be a hypergraph of maximal vertex degree $Δ$, such that each its hyperedge contains at least $δ$ vertices. Let $k=\lceil\frac{2Δ}δ\rceil$. We prove that (i) The hypergraph $\mathcal{H}$ admits proper vertex coloring in $k+1$ colors. (ii) The hypergraph $\mathcal{H}$ admits proper vertex coloring in $k$ colors, if $δ\ge 3$ and $k\ge 3$. As a consequence of these results we derive upper bounds on the number of colors in dynamic colorings.