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 ℝ3 whose intersection graph is triangle-free of chromatic number Ω (n1∕15). This improves the previously best known bound Ω (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 ℝd, whose Delaunay graph with respect to axis-parallel boxes has independence number at most n⋅(log n)−(𝑑−1)∕2+o(1). This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.