{
  "abstract": "Canonical Ramsey theory studies structured colour patterns in colourings with arbitrarily many colours. Lefmann's canonical Rado theorem extends the canonical van der Waerden theorem of Erdős and Graham to partition-regular linear systems. Recently, Alvarado, Kohayakawa, Morris, Mota and Ortega proved a sparse canonical van der Waerden theorem for random set $[n]_p$. In this paper, we prove a sparse canonical Rado theorem. For every irredundant partition-regular matrix $A$ of positive rank with $k\\ge3$ columns, and every prescribed partition witnessing the columns condition, if $p\\ge Cn^{-1/m(A)}$, then with probability at least $1-e^{-cnp}$ every colouring of $[n]_p$ contains $$Ω\\big(p^k n^{k-\\rank(A)}\\big)$$ canonical solutions of $A\\mathbf x=0$ with distinct coordinates. A matching 0-statement holds when the prescribed partition has one or at least three blocks; the Schur equation shows that it can fail for two-block patterns. For every prescribed partition, we also establish a robust threshold at the same scale under deletion of a fixed positive proportion of the random set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37053",
  "author_total": 4,
  "authors": [
    {
      "name": "Zhenzhen Lou"
    },
    {
      "name": "Guangfu Wang"
    },
    {
      "name": "Man Wu"
    },
    {
      "name": "Zhifei Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5p2vjp02pts4a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37053",
  "fulltext_url": null,
  "id": "5p2vjp02pts4a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37053",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7224124594",
  "pdf_url": "https://arxiv.org/pdf/2610.37053",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5p2vjp02pts4a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37053",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37053"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A sparse canonical Rado theorem",
  "title_text": "A sparse canonical Rado theorem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5p2vjp02pts4a",
  "venue": "arXiv",
  "versions": []
}