Random coloring method in the combinatorial problem of Erdős and Lovász
Random Structures & Algorithms
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A random construction gives new examples of simple hypergraphs with high chromatic number that have few edges and-or low maximum degree. In particular, for every integers k ≥ 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 kr-1 lnk⌉. This is only 4r2 lnk times greater than the lower bound by Erdős and Lovász (Colloquia Math Soc János Bolyai 10 (1973), 609–627) that holds for all hypergraphs (without restrictions on girth). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010