Abstract
AbstractThe 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 lines in whose intersection graph is triangle‐free of chromatic number . This improves the previously best known bound 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 points in , whose Delaunay graph with respect to axis‐parallel boxes has independence number at most . This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.
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