Computational Geometry 20 JAN, 2015

A note on coloring line arrangements

A note on coloring line arrangements

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.

💡 Research Summary

The paper addresses the problem of coloring the lines of a planar arrangement so that no face (i.e., the polygonal region bounded by the lines) is monochromatic. Given an arrangement of n lines in general position, the authors prove that O(√(n/ log n)) colors are sufficient to achieve a non‑monochromatic coloring of every face. This improves the previous bound of O(√n) established by Bose, Choudhary, Cox, and Kumar (2012) by a factor of Θ(√log n).

Background and Motivation
A line arrangement partitions the plane into O(n²) faces. A coloring of the lines induces a coloring of each face by the set of colors of the lines that bound it. The goal is to avoid any face whose incident lines all share the same color. This problem is a geometric analogue of graph coloring and has connections to combinatorial geometry, particularly to a longstanding question of Erdős concerning the maximum size of a subset in general position within a set of n points that contains no four collinear points. Improving the line‑coloring bound directly translates into stronger lower bounds for that Erdős problem.

Main Result
Theorem. For every arrangement of n lines in the plane, there exists a coloring with k = O(√(n/ log n)) colors such that every face contains at least two distinct colors.

Proof Sketch

  1. Partition of Lines. The n lines are divided into t = ⌈c·√(n log n)⌉ groups (for a sufficiently large constant c). Each group has size Θ(√(n log n)).
  2. Random Assignment. Each group receives a distinct color; within a group all lines share that color. This yields a random coloring once the grouping is chosen uniformly at random.
  3. Bad Events. For each face f define the bad event A_f = “all lines incident to f belong to the same group (hence have the same color).”
  4. Probability Estimate. A face is bounded by O(√n) lines. The probability that all these lines fall into the same group is at most (1/t)^{2} = O(1/(n log n)). Hence p = Pr