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 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.

中文翻译：

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相对于框的着色线和 Delaunay 图

本文的目标是证明（使用概率工具）具有某些有趣的组合属性的线、框和点的配置的存在。(i) 首先，我们构建一个家庭行中其交图是无色数三角形。这改善了以前最知名的界限由 Norin 提出，也是第一个构造具有多项式色数的简单几何对象的无三角形交集图。(ii) 其次，我们构建了一组点在，其关于轴平行框的 Delaunay 图最多具有独立数。这扩展了 Chen、Pach、Szegedy 和 Tardos 考虑的平面情况。