{
  "abstract": "For integers $n,r,t$ with $2\\le t\\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determine this function for every $n\\ge1$. First, the corresponding three-colourable extremum $δ(n,7,3)$ is reduced to a $7\\times3$ integer matrix problem and determined exactly. Second, a structural argument shows that every balanced $7$-partite $K_4$-free graph $G$ with $δ(G)>\\frac{132}{31}n$ is three-colourable. Consequently, $f(n,7,4)=\\lfloor30n/7\\rfloor$ except when $n\\equiv4\\pmod7$ and $n\\ge11$, where it is one less.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19177",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuuki Kasugai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cschg0c43efw2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19177",
  "fulltext_url": null,
  "id": "cschg0c43efw2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19177",
  "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": "W7213602868",
  "pdf_url": "https://arxiv.org/pdf/2609.19177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cschg0c43efw2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Remaining $K_4$-Free Case in the Multipartite Clique Problem",
  "title_text": "The Remaining K_4-Free Case in the Multipartite Clique Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/cschg0c43efw2",
  "venue": "arXiv",
  "versions": []
}