完整简单拓扑图中不可避免的模式

Pub Date : 2024-05-29 DOI:10.1007/s00454-024-00658-6

Andrew Suk, Ji Zeng

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引用次数: 0

摘要

我们证明了每一个完整的 n 个顶点的简单拓扑图都包含一个至少 \((\log n)^{1/4 - o(1)}\) 个顶点的拓扑子图，它与完整的凸几何图或完整的扭曲图具有弱同构性。这是对 Pach、Solymosi 和 Tóth 于 2003 年得到的边界 \(\Omega (\log ^{1/8}n)\) 的首次改进。我们还证明了每一个完整的 n 顶点简单拓扑图都包含一条长度至少为 \((\log n)^{1 -o(1)}\) 的平面路径。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Unavoidable Patterns in Complete Simple Topological Graphs

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least \((\log n)^{1/4 - o(1)}\) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound \(\Omega (\log ^{1/8}n)\) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least \((\log n)^{1 -o(1)}\).

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