{"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}
引用次数: 0
Abstract
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 $G$.