{"title":"将 Borel 图嵌入近似最优维度的网格中","authors":"Anton Bernshteyn, Jing Yu","doi":"arxiv-2407.19785","DOIUrl":null,"url":null,"abstract":"Let $G$ be a Borel graph all of whose finite subgraphs embed into the\n$d$-dimensional grid with diagonals. We show that then $G$ itself admits a\nBorel embedding into the Schreier graph of a free Borel action of $\\mathbb\nZ^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$\nis replaced by $O(\\rho \\log \\rho)$, where $\\rho$ is the polynomial growth rate\nof $G$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Embedding Borel graphs into grids of asymptotically optimal dimension\",\"authors\":\"Anton Bernshteyn, Jing Yu\",\"doi\":\"arxiv-2407.19785\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $G$ be a Borel graph all of whose finite subgraphs embed into the\\n$d$-dimensional grid with diagonals. We show that then $G$ itself admits a\\nBorel embedding into the Schreier graph of a free Borel action of $\\\\mathbb\\nZ^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$\\nis replaced by $O(\\\\rho \\\\log \\\\rho)$, where $\\\\rho$ is the polynomial growth rate\\nof $G$.\",\"PeriodicalId\":501444,\"journal\":{\"name\":\"arXiv - MATH - Metric Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Metric Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.19785\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.19785","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Embedding Borel graphs into grids of asymptotically optimal dimension
Let $G$ be a Borel graph all of whose finite subgraphs embed into the
$d$-dimensional grid with diagonals. We show that then $G$ itself admits a
Borel embedding into the Schreier graph of a free Borel action of $\mathbb
Z^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$
is replaced by $O(\rho \log \rho)$, where $\rho$ is the polynomial growth rate
of $G$.