{
  "abstract": "We consider infinite walks in $\\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \\in \\{(6,3), (4,4), (3,7)\\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also imply the existence of infinite words over small finite alphabets that are weakly abelian squarefree (resp., weakly abelian cubefree, weakly abelian 6th-power-free).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20366",
  "author_total": 3,
  "authors": [
    {
      "name": "Stijn Cambie"
    },
    {
      "name": "Erik Kalviainen"
    },
    {
      "name": "J. Shallit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aevcv1qdwx12x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20366",
  "fulltext_url": null,
  "id": "aevcv1qdwx12x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20366",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213606025",
  "pdf_url": "https://arxiv.org/pdf/2609.20366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aevcv1qdwx12x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Brown-Gerver-Ramsey Theorems in Small Dimensions",
  "title_text": "Brown-Gerver-Ramsey Theorems in Small Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/aevcv1qdwx12x",
  "venue": "arXiv",
  "versions": []
}