{
  "abstract": "Löfberg and Qi introduced the second order Zarankiewicz number \\(z_2\\), the recursive-line Zarankiewicz number \\(z_{RL}\\), and the signed Zarankiewicz number \\(z_{SL}\\) for doubly simple biquadratic forms. It was shown that \\[ z_2(m,n)\\ge z_{SL}(m,n)\\ge z_{RL}(m,n) \\] for all \\(m\\) and \\(n\\). However, there was no evidence that there exist particular \\(m\\) and \\(n\\) such that \\(z_{SL}(m,n)>z_{RL}(m,n)\\). The motivation for introducing \\(z_{SL}\\) was as follows: during the study of the exceptional case \\(m=15\\), \\(n=6\\), Löfberg and Qi showed that \\[ z_2(15,6)=z_{SL}(15,6)=60, \\] but the exact value of \\(z_{RL}(15,6)\\) was unknown then. In this paper we show that \\[ z_{RL}(15,6)=60. \\] This eliminates the motivation for introducing \\(z_{SL}\\). Whether \\(z_{SL}(m,n)=z_{RL}(m,n)\\) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number \\(z_A\\) from the limited augmented Zarankiewicz number \\(z_L\\) at \\(m=n=1893\\). We hope that the separation problem for \\(z_{SL}\\) and \\(z_{RL}\\) can also be solved. We also present the exact values of \\(z_{RL}(m,6)\\) for \\(6\\le m\\le 16\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20071",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhiwei Chen"
    },
    {
      "name": "Yannan Chen"
    },
    {
      "name": "Liqun Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ggfjf4tm4vhp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20071",
  "fulltext_url": null,
  "id": "9ggfjf4tm4vhp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20071",
  "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": "W7213583285",
  "pdf_url": "https://arxiv.org/pdf/2609.20071",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ggfjf4tm4vhp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20071",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20071"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?",
  "title_text": "Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/9ggfjf4tm4vhp",
  "venue": "arXiv",
  "versions": []
}