Coloring lines and Delaunay graphs with respect to boxes

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 $\mathrm{\Omega }(n^{1\mathrm{/}15})$. This improves the previously best known bound $\mathrm{\Omega }(\log \log n)$ 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 $n\cdot (\log n{)}^{-(d-1)\mathrm{/}2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.