{
  "abstract": "The Borel distinguishing number $D_B(\\mathcal{G})$ of a Borel graph $\\mathcal{G}$, recently introduced by Bilge and Kaya, is the minimum number of colors required to break the symmetry of $\\mathcal{G}$ in a Borel way. In this paper, we investigate the Borel distinguishing number of Schreier graphs induced by the free part of the shift action $Γ\\curvearrowright n^Γ$. We prove that $D_B(\\mathcal{G})\\le n+1$ for $Γ=\\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\\mathcal{G})\\ge n+1$ if $ Γ$ is amenable and $ \\{γ\\in \\mathrm{Aut}(\\mathrm{Cay}(Γ,S)) \\mid γ(e) = e \\}$ is non-trivial. We also show that $D_B(\\mathcal{G})$ is finite if $Γ$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15003",
  "author_total": 2,
  "authors": [
    {
      "name": "Junhao Chen"
    },
    {
      "name": "Jie Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bt1maefwea13.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15003",
  "fulltext_url": null,
  "id": "9bt1maefwea13",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15003",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7213292561",
  "pdf_url": "https://arxiv.org/pdf/2609.15003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bt1maefwea13.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The Borel Distinguishing Number of Schreier Graphs",
  "title_text": "The Borel Distinguishing Number of Schreier Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9bt1maefwea13",
  "venue": "arXiv",
  "versions": []
}