ABSTRACT:We show that for all large n, every n-uniform hypergraph with at most 0 7 n/ ln n × 2 n edges can be 2-colored. This makes progress on a problem of Erdős [Nordisk Mat. Tidskrift 11, 5-10 (1963)], improving the previous-best bound of n 1/3−o 1 × 2 n due to Beck [Discrete Math. 24, 127-137 (1978)]. We further generalize this to a "local" version, improving on one of the first applications of the Lovász local lemma. We also present fast randomized algorithms that output a proper 2-coloring with high probability for n-uniform hypergraphs with at most 0 7 n/ ln n × 2 n edges, for all large n. In addition, we derandomize and parallelize these algorithms, to derive NC 1 versions of these results.

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