Coloring lines and Delaunay graphs with respect to boxes

TL;DR

This paper demonstrates the existence of geometric configurations with high chromatic number and small independence number, using probabilistic methods to improve bounds in intersection and Delaunay graphs of lines, points, and boxes.

Contribution

It provides new constructions of geometric graphs with polynomial chromatic number and small independence number, improving previous bounds and extending known results to higher dimensions.

Findings

Constructed a family of lines with triangle-free intersection graph and high chromatic number.

Extended bounds on Delaunay graphs with respect to boxes to higher dimensions.

Improved the known lower bounds for chromatic number in geometric intersection graphs.

Abstract

The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $\mathbb{R}^3$ whose intersection graph is triangle-free of chromatic number $\Omega(n^{1/15})$. This improves the previously best known bound $\Omega(\log\log n)$ by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of $n$ points in $\mathbb{R}^d$, whose Delaunay graph with respect to axis-parallel boxes has independence number at most $n\cdot (\log n)^{-(d-1)/2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.