A note on two-colorability of nonuniform hypergraphs

03/08/2018

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by   Lech Duraj, et al.

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For a hypergraph H, let q(H) denote the expected number of monochromatic edges when the color of each vertex in H is sampled uniformly at random from the set of size 2. Let s_(H) denote the minimum size of an edge in H. Erdős asked in 1963 whether there exists an unbounded function g(k) such that any hypergraph H with s_(H) ≥ k and q(H) ≤ g(k) is two colorable. Beck in 1978 answered this question in the affirmative for a function g(k) = Θ(^* k). We improve this result by showing that, for an absolute constant δ>0, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph H with s_(H) ≥ k and q(H) ≤δ· k.