Quasi-treeings are treeable: a streamlined proof
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910917546999808 |
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| author | Zhai, Zhaoshen |
| author_facet | Zhai, Zhaoshen |
| contents | We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' replace each component of the graph by a tree of special ultrafilter-like objects on cuts called orientations; moreover, if the cuts are dense towards ends, then the union of these trees is a Borel treeing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-treeings are treeable: a streamlined proof Zhai, Zhaoshen Logic Combinatorics 03E15, 20F65, 20E08, 37A20 We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' replace each component of the graph by a tree of special ultrafilter-like objects on cuts called orientations; moreover, if the cuts are dense towards ends, then the union of these trees is a Borel treeing. |
| title | Quasi-treeings are treeable: a streamlined proof |
| topic | Logic Combinatorics 03E15, 20F65, 20E08, 37A20 |
| url | https://arxiv.org/abs/2409.09843 |