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Eyal Ackerman
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János Pach
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Rom Pinchasi
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Radoš Radoičić
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Géza Tóth
Keywords:
line arrangement, chromatic number, coloring, hypergraphs
Abstract
We show that the lines of every arrangement of $n$ lines in the plane can be colored with $O(\sqrt{n/ \log n})$ colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. by a $\Theta(\sqrt{\log n})$ factor. Any further improvement on this bound would also improve the best known lower bound on the following problem of Erdős: estimate the maximum number of points in general position within a set of $n$ points containing no four collinear points.