少量点上简单复合体的嵌入维数

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2023-03-28 DOI:10.1007/s00026-023-00644-4

Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon

{"title":"少量点上简单复合体的嵌入维数","authors":"Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon","doi":"10.1007/s00026-023-00644-4","DOIUrl":null,"url":null,"abstract":"<div><p>We provide a simple characterization of simplicial complexes on few vertices that embed into the <i>d</i>-sphere. Namely, a simplicial complex on <span>\\(d+3\\)</span> vertices embeds into the <i>d</i>-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 4","pages":"993 - 1003"},"PeriodicalIF":0.6000,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00644-4.pdf","citationCount":"3","resultStr":"{\"title\":\"Embedding Dimensions of Simplicial Complexes on Few Vertices\",\"authors\":\"Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon\",\"doi\":\"10.1007/s00026-023-00644-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We provide a simple characterization of simplicial complexes on few vertices that embed into the <i>d</i>-sphere. Namely, a simplicial complex on <span>\\\\(d+3\\\\)</span> vertices embeds into the <i>d</i>-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.</p></div>\",\"PeriodicalId\":50769,\"journal\":{\"name\":\"Annals of Combinatorics\",\"volume\":\"27 4\",\"pages\":\"993 - 1003\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-03-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00026-023-00644-4.pdf\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00026-023-00644-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00644-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 3

摘要

我们提供了嵌入d球面的几个顶点上的单纯复形的一个简单表征。也就是说，在\（d+3\）顶点上的单纯复形嵌入到d球面中，当且仅当其非面不形成相交族。作为直接结果，我们恢复了经典的van Kampen–Flores定理，并提供了Erdõs–Ko–Rado定理的拓扑扩展。通过与平面图的Fáry定理的类比，我们还证明了这种复形满足连续可嵌入性和线性可嵌入性等价的刚性性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Embedding Dimensions of Simplicial Complexes on Few Vertices

We provide a simple characterization of simplicial complexes on few vertices that embed into the d-sphere. Namely, a simplicial complex on \(d+3\) vertices embeds into the d-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches