Constructions of sparse uniform hypergraphs with high chromatic number

A random construction gives new examples of simple hyper graphs with high chromatic number that have few edges and/ or low maximum degree. In particular, for every integer sk ≥ 2,r ≥ 2, and g ≥ 3, there exist r-uniform non-k-colorable hypergraphs of girth at least g with maximum degree at most ⌈r k^r-1 ln k⌉. This is only 4r² ln k times greater than the lower bound by Erdos and Lovász (Colloquia Math Soc János Bolyai 10 (1973), 609-627) that holds for all hypergraphs (without restrictions on girth).