Improved bounds and algorithms for hypergraph two-coloring

We show that for all large n, every n-uniform hypergraph with at most 0.7√(n/lnn)×2ⁿ edges can be two-colored. We, in fact, present fast algorithms that output a proper two-coloring with high probability for such hypergraphs. We also derandomize and parallelize these algorithms, to derive NC¹ versions of these results. This makes progress on a problem of Erdos (1963), improving the previous-best bound of n^1/3-0(1)×2ⁿ due to Beck (1978). We further generalize this to a “local” version, improving on one of the first applications of the Lovasz Local Lemma