{
  "abstract": "Let $G$ be a $3$-partite graph with $k$ vertices in each part such that the bipartite graph induced by any two parts contains no cycle of length four. Fischer and Matoušek [J. Combin. Theory Ser. A, 2001] asked for the maximum number of triangles in such a graph. They obtained the lower bound $(1-o(1))k^{3/2}$ and the upper bound $k^{7/4}+O(k^{3/2})$. Coulter, Matthews and Timmons [J. Combin. Theory Ser. B, 2018] later constructed such graphs using planar polynomials over finite fields and improved the lower bound to $(1-o(1))k^{5/3}$. In this note, we use a new triple of planar polynomials and further improve the lower bound to $(1-o(1))k^{17/10}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06594",
  "author_total": 2,
  "authors": [
    {
      "name": "Chunqiu Fang"
    },
    {
      "name": "Rongxing Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kr3ntqa74fkj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06594",
  "fulltext_url": null,
  "id": "5kr3ntqa74fkj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06594",
  "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": "W7212087144",
  "pdf_url": "https://arxiv.org/pdf/2609.06594",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kr3ntqa74fkj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06594",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the maximum number of triangles in tripartite graphs with no $4$-cycles between any two parts",
  "title_text": "On the maximum number of triangles in tripartite graphs with no $4$-cycles between any two parts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5kr3ntqa74fkj",
  "venue": "arXiv",
  "versions": []
}