{
  "abstract": "For positive integers $n$ and $d$, the $d$-dimensional grid of side length $n$ is the graph $G(n,d)$ with vertex set $[n]^d$, in which two vertices are adjacent if they differ by $1$ in exactly one coordinate and agree in all the other coordinates. We determine, for every fixed $d$, the asymptotic growth in $r$ of the $r$th strong coloring numbers of these grids. More precisely for $n$ large enough, ${\\rm scol}_r(G(n,d)) = Ω_d\\bigl(r^{d-1}\\bigr)$ as $r\\to\\infty$. This matches an upper bound given by a volume packing argument, which holds more generally for touching graphs of unit balls in $\\mathbb R^d$. For the lower bound, we construct in $G(n,d)$ a bramble of order $Ω_d(n^{d-1})$ with each element of radius at most $dn$. This construction was found by OpenAI's GPT-5.6 Sol.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21506",
  "author_total": 2,
  "authors": [
    {
      "name": "Piotr Micek"
    },
    {
      "name": "Torsten Ueckerdt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/326jfnmevgrk4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21506",
  "fulltext_url": null,
  "id": "326jfnmevgrk4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21506",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222610787",
  "pdf_url": "https://arxiv.org/pdf/2610.21506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/326jfnmevgrk4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Strong coloring numbers of grids",
  "title_text": "Strong coloring numbers of grids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/326jfnmevgrk4",
  "venue": "arXiv",
  "versions": []
}