{
  "abstract": "Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $κ$-colored sequence of $n \\ge d+1$ points in general position in $\\mathbb{R}^d$. Let $ρ$ be a $κ$-colored order type on $k \\le d+1$ points that has positive density on $P$; that is, for some constant $δ>0$, there are $δ\\cdot \\binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $ρ$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, δ$, $k$ and $κ$) and disjoint subsets $X_1,\\dots,X_k$ of $P$, each with at least $c \\cdot n$ points, such that for every choice of $k$ points $x_i \\in X_i$, $(x_1,\\dots,x_k)$ has the same order type and color pattern as $ρ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.07806",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruy Fabila-Monroy"
    },
    {
      "name": "Benedikt Hahn"
    },
    {
      "name": "Jesús Leaños"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gexg8myzmjfr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.07806",
  "fulltext_url": null,
  "id": "7gexg8myzmjfr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.07806",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7164020166",
  "pdf_url": "https://arxiv.org/pdf/2606.07806",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gexg8myzmjfr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2606.07806",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.07806"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Blow-ups of order types of positive density",
  "title_text": "Blow-ups of order types of positive density",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/7gexg8myzmjfr",
  "venue": "arXiv",
  "versions": []
}