Measurable domatic partitions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911207794933760 |
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| author | Hou, Edward |
| author_facet | Hou, Edward |
| contents | Let $Γ$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq Γ$, a domatic $\aleph_0$-partition (for its Schreier graph on $Γ$) is a partial function $f:Γ\rightharpoonup\mathbb{N}$ such that for every $x\in Γ$, one has $f[S\cdot x]=\mathbb{N}$. We show that a continuous domatic $\aleph_0$-partition exists, if and only if a Baire measurable domatic $\aleph_0$-partition exists, if and only if the topological closure of $S$ is uncountable. A Haar measurable domatic $\aleph_0$-partition exists for all choices of $S$. We also investigate domatic partitions in the general descriptive graph combinatorial setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05751 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Measurable domatic partitions Hou, Edward Logic Combinatorics Let $Γ$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq Γ$, a domatic $\aleph_0$-partition (for its Schreier graph on $Γ$) is a partial function $f:Γ\rightharpoonup\mathbb{N}$ such that for every $x\in Γ$, one has $f[S\cdot x]=\mathbb{N}$. We show that a continuous domatic $\aleph_0$-partition exists, if and only if a Baire measurable domatic $\aleph_0$-partition exists, if and only if the topological closure of $S$ is uncountable. A Haar measurable domatic $\aleph_0$-partition exists for all choices of $S$. We also investigate domatic partitions in the general descriptive graph combinatorial setting. |
| title | Measurable domatic partitions |
| topic | Logic Combinatorics |
| url | https://arxiv.org/abs/2205.05751 |