  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.
