Multipass greedy coloring of simple uniform hypergraphs

Let m∗(n) be the minimum number of edges in an n-uniform simple hypergraph that is not two colorable. We prove that m∗(n) = Ω(4n/ ln(n)). Our result generalizes to r-coloring of b-simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b-simple n-uniform hypergraph that is not r-colorable must be Ω(rn/ ln(n)). By trimming arguments it implies that every such graph has Ω((rn/ ln(n))b+1/b) edges. For any fixed r > 2 our techniques yield also a lower bound Ω(rn/ ln(n)) for van der Waerden numbers W (n, r).