Embedding Borel graphs into grids of asymptotically optimal dimension
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866912769671954432 |
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| author | Bernshteyn, Anton Yu, Jing |
| author_facet | Bernshteyn, Anton Yu, Jing |
| contents | 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 $O(ρ\log ρ)$, where $ρ$ is the polynomial growth rate of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embedding Borel graphs into grids of asymptotically optimal dimension Bernshteyn, Anton Yu, Jing Combinatorics Logic Metric Geometry 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 $O(ρ\log ρ)$, where $ρ$ is the polynomial growth rate of $G$. |
| title | Embedding Borel graphs into grids of asymptotically optimal dimension |
| topic | Combinatorics Logic Metric Geometry |
| url | https://arxiv.org/abs/2407.19785 |