Geometric constructions: lines and boxes

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 R 3 whose intersection graph is triangle-free of chromatic number Ω(n 1/15 ). 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) Furthermore, we construct a set of n points P and a family of n axis-parallel boxes B in R d such that the incidence graph of (P, B) is K 2,2 -free of average degree at least (log n) d−1−o(1) . This also provides an n-vertex graph of separation dimension 2d and average degree (log n) d−1−o(1) . The former answers a question of Basit, Chernikov, Starchenko, Tao, and Tran, while the latter addresses a problem of Alon, Basavaraju, Chandran, Mathew, and Rajendraprasad.(iii) Finally, we construct a set of n points in R d , 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.