{
  "abstract": "Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \\$500 prize from Erdős. In the 1980s, Erdős and Sós asked to determine Turán densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Turán density of $K_4^{(3)}$ is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08336",
  "author_total": 6,
  "authors": [
    {
      "name": "Bartłomiej Kielak"
    },
    {
      "name": "Daniel Král"
    },
    {
      "name": "Ander Lamaison"
    },
    {
      "name": "Hong Liu"
    },
    {
      "name": "Xichao Shu"
    },
    {
      "name": "Zhuo Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2eztk4jxh1x4c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08336",
  "fulltext_url": null,
  "id": "2eztk4jxh1x4c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08336",
  "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": "W7212033496",
  "pdf_url": "https://arxiv.org/pdf/2609.08336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2eztk4jxh1x4c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.08336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08336"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Solution of uniform Turán's Tetrahedron Problem",
  "title_text": "Solution of uniform Turán's Tetrahedron Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2eztk4jxh1x4c",
  "venue": "arXiv",
  "versions": []
}