{
  "abstract": "A partial matrix has entries in $\\{0,1,\\star\\}$, and a disambiguation replaces each $\\star$ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For $0<\\varepsilon<π/2$, let $M_\\varepsilon^d$ be the partial matrix indexed by points of the unit sphere $\\mathbb S^d$, with entry $0$ for pairs at spherical distance at most $\\varepsilon$, $1$ for pairs at distance at least $π-\\varepsilon$, and $\\star$ otherwise. Although these matrices have VC-dimension bounded independently of $d$, we prove that every disambiguation contains every binary $k\\times k$ matrix once $d$ is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every $k$, a finite partial matrix whose fully specified $2\\times2$ submatrices are all constant, while every disambiguation contains every binary $k\\times k$ matrix. A symmetric analogue holds for partial graphs: for every $k$, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every $k$-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16359",
  "author_total": 3,
  "authors": [
    {
      "name": "Romain Bourneuf"
    },
    {
      "name": "Antonin Kiladjian"
    },
    {
      "name": "Stéphan Thomassé"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcyhd6r8mr9an.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16359",
  "fulltext_url": null,
  "id": "fcyhd6r8mr9an",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16359",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213404931",
  "pdf_url": "https://arxiv.org/pdf/2609.16359",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcyhd6r8mr9an.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16359",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16359"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ramsey Obstructions to Disambiguation",
  "title_text": "Ramsey Obstructions to Disambiguation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/fcyhd6r8mr9an",
  "venue": "arXiv",
  "versions": []
}