{
  "abstract": "Classical Turán theory relates the asymptotic extremal problem for an ordinary forbidden graph to its chromatic number. This principle fails when a forbidden copy must respect prescribed vertex colors, as occurs naturally in multipartite extremal problems and in embeddings with prescribed clusters. We study properly vertex-colored graphs whose hosts have a prescribed density between every pair of color classes, with containment interpreted through color-preserving homomorphisms. Our main theorem gives a finite directed reduction: for every fixed $m$ and $\\varepsilon>0$, every sufficiently many-colored $d$-dense host admits a colored homomorphism from an $m$-layer half-blowup $D[m]$ of a finite digraph $D$ with density at least $d-\\varepsilon$, where $D$ may be chosen with $\\lceil\\varepsilon^{-1}\\rceil$ vertices. Consequently, the colored Turán density of every non-empty finite forbidden family equals the homomorphism Turán density of its associated acyclic orientations. We then determine the complete finite value set: these densities are exactly the Lagrangians of finite loopless digraphs, and hence are rational. The main technical input is a low-resolution sampling method that preserves all pairwise bipartite degree profiles on a large subfamily of color classes while keeping the sample size independent of the number of retained classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17992",
  "author_total": 4,
  "authors": [
    {
      "name": "Nannan Chen"
    },
    {
      "name": "Ander Lamaison"
    },
    {
      "name": "Heng Li"
    },
    {
      "name": "Hong Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2wyb14frfptvw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17992",
  "fulltext_url": null,
  "id": "2wyb14frfptvw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17992",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221754378",
  "pdf_url": "https://arxiv.org/pdf/2610.17992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2wyb14frfptvw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Colored Turán densities and finite directed templates",
  "title_text": "Colored Turán densities and finite directed templates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2wyb14frfptvw",
  "venue": "arXiv",
  "versions": []
}