{
  "abstract": "For positive integers $n,r,t$, let $δ(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determine the corresponding multipartite clique threshold up to an additive constant in a broad parameter range. I determine the chromatic parameter throughout the range $r=mt-a$, $m\\ge2$, $t\\ge3$, $2\\le a\\le \\min\\{m,t-1\\}$. The answer differs from the Lo--Treglown--Zhao upper bound by at most one. I give an explicit arithmetic criterion deciding when this one-unit correction occurs. The proof reduces the problem to an integer matrix extremum. In the boundary case, equality forces the supports of all mixed rows to form a spanning star, after which the only remaining obstruction is a divisibility condition. Combining this formula with the Andrasfai--Erdos--Sos theorem sharpens the known equality range for $f(n,r,t+1)=δ(n,r,t)$. In particular, for $t=3$ it removes the remaining size restrictions at $r=10$ and $r=13$. Together with the $r=7$ result in arXiv:2609.19177, the classical $r=4$ case, and the known congruence classes, this gives a formula for the multipartite $K_4$-free problem for every admissible $r\\ge4$ and every $n\\ge1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27503",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuuki Kasugai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ptw8a4xhsb67.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27503",
  "fulltext_url": null,
  "id": "4ptw8a4xhsb67",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27503",
  "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": "W7214150608",
  "pdf_url": "https://arxiv.org/pdf/2609.27503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ptw8a4xhsb67.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.27503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Chromatic Extremal Thresholds and the Multipartite $K_4$-Free Problem",
  "title_text": "Chromatic Extremal Thresholds and the Multipartite K_4-Free Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/4ptw8a4xhsb67",
  "venue": "arXiv",
  "versions": []
}