We show that any set of n points in general position in the plane determines $n^{1-o(1)}$ pairwise crossing segments. The best previously known lower bound, $\mathrm{\Omega }\left(\sqrt{n}\right)$, was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

中文翻译：

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平面点集确定许多成对的交叉线段

我们表明，平面中一般位置的任何一组n个点决定$n^{1-○(1)}$成对交叉段。以前已知的最好的下界，$\mathrm{\Omega }\left(\sqrt{n}\right)$25 多年前，Aronov、Erdős、Goddard、Kleitman、Klugerman、Pach 和 Schulman 证明了这一点。我们的证明是完全建设性的，并扩展到密集的几何图形。