A note on coloring line arrangements

TL;DR

This paper improves the upper bound on coloring lines in arrangements to prevent monochromatic faces, linking this geometric problem to a longstanding question in combinatorial geometry about point sets with no four collinear points.

Contribution

It provides a tighter upper bound of O(√(n/ log n)) colors for line arrangements, advancing the understanding of geometric coloring problems.

Findings

Bound of O(√(n/ log n)) colors for line arrangements

Connection between coloring arrangements and Erdős's problem

Improvement over previous bounds by a logarithmic factor

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. \cite{BCC12} by a $\Theta(\sqrt{\log n})$ factor. Any further improvement on this bound will improve the best known lower bound on the following problem of Erd\H{o}s: Estimate the maximum number of points in general position within a set of $n$ points containing no four collinear points.