On ends of degree $ω_1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910689532051456 |
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| author | Aurichi, Leandro Fernandes, Gabriel Júnior, Paulo Magalhães |
| author_facet | Aurichi, Leandro Fernandes, Gabriel Júnior, Paulo Magalhães |
| contents | We prove that if $ T $ is a semi-special tree that is not special, then there exists a graph $ G $, formed as an inflation of a sparse $ T $-graph, such that for any special tree $ S $, $ G $ is not a subdivision of an inflation of an sparse $ S $-graph. Furthermore $G$ has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin's grid theorem, extending it to ends of degree $ \aleph_1 $, which was originally established for ends of countable degree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On ends of degree $ω_1$ Aurichi, Leandro Fernandes, Gabriel Júnior, Paulo Magalhães Logic Combinatorics 05C63, 03E05 We prove that if $ T $ is a semi-special tree that is not special, then there exists a graph $ G $, formed as an inflation of a sparse $ T $-graph, such that for any special tree $ S $, $ G $ is not a subdivision of an inflation of an sparse $ S $-graph. Furthermore $G$ has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin's grid theorem, extending it to ends of degree $ \aleph_1 $, which was originally established for ends of countable degree. |
| title | On ends of degree $ω_1$ |
| topic | Logic Combinatorics 05C63, 03E05 |
| url | https://arxiv.org/abs/2411.05241 |