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. Erdős and Lovász proved that m(r, t, 3) ≥ t 2(r−2) 16r(r − 1)2 and m(r, t, g) ≤ 4 · 20g−1r3g−5t(g−1)(r+1). A result of Szabó improves the lower bound by a factor of r2− 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 > 0 and sufficiently large r, every r-uniform b-simple hypergraph H with maximum edge-degree at most trr1− is t-colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 35, 348–368, 2009