{
  "abstract": "Balogh, Clemen, and Lidický proved that the $\\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.02210",
  "author_total": 5,
  "authors": [
    {
      "name": "Wanfang Chen"
    },
    {
      "name": "Jinghua Deng"
    },
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Xizhi Liu"
    },
    {
      "name": "Yixiao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/em14hemhq9pkz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.02210",
  "fulltext_url": null,
  "id": "em14hemhq9pkz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.02210",
  "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": "W7163231726",
  "pdf_url": "https://arxiv.org/pdf/2606.02210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/em14hemhq9pkz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.02210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.02210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Vertex-colored Turán theorems with applications in extremal hypergraph problems",
  "title_text": "Vertex-colored Turán theorems with applications in extremal hypergraph problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/em14hemhq9pkz",
  "venue": "arXiv",
  "versions": []
}