{"title":"完整简单拓扑图中不可避免的模式","authors":"Andrew Suk, Ji Zeng","doi":"10.1007/s00454-024-00658-6","DOIUrl":null,"url":null,"abstract":"<p>We show that every complete <i>n</i>-vertex simple topological graph contains a topological subgraph on at least <span>\\((\\log n)^{1/4 - o(1)}\\)</span> vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound <span>\\(\\Omega (\\log ^{1/8}n)\\)</span> obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete <i>n</i>-vertex simple topological graph contains a plane path of length at least <span>\\((\\log n)^{1 -o(1)}\\)</span>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unavoidable Patterns in Complete Simple Topological Graphs\",\"authors\":\"Andrew Suk, Ji Zeng\",\"doi\":\"10.1007/s00454-024-00658-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that every complete <i>n</i>-vertex simple topological graph contains a topological subgraph on at least <span>\\\\((\\\\log n)^{1/4 - o(1)}\\\\)</span> vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound <span>\\\\(\\\\Omega (\\\\log ^{1/8}n)\\\\)</span> obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete <i>n</i>-vertex simple topological graph contains a plane path of length at least <span>\\\\((\\\\log n)^{1 -o(1)}\\\\)</span>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00658-6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00658-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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。
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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