Coloring uniform hypergraphs with few edges

A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m(r,t,g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g. Erdo{double acute}s and Lovász proved that A result of Szabó improves the lower bound by a factor of r^2-ε for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b-simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b,t, and ε > 0 and sufficiently large r, every r-uniform b-simple hypergraph H with maximum edge-degree at most t^rr^1-ε is t-colorable. Some results hold for list coloring, as well.