{
  "abstract": "For $n\\ge6$, $m=\\binom n2$, let the complete-graph incidence family on $K_n$ have the vertices of $K_n$ as columns, its edges as rows, and the incidence graph as one-edge graph. The universal cell bound of Löfberg and Qi gives $z_2(m,n)\\le Z(n):=\\lfloor n(n-1)(n+2)/4\\rfloor$. We implement the nested one-factorization construction of that family and determine exactly what it certifies. For $n=2q$ with $q$ an odd prime, $q\\ge5$, the construction has no hole and the cross-factor transfer equations apply verbatim, giving $z_2=z_{SL}=z_{RL}=Z(n)$; $n=6$ is settled by a separate cyclic witness. For odd $n=2p+1$ the near-perfect one-factorization yields Hamilton paths closed into odd cycles, and the transfer argument breaks: that scheme certifies only $R(G_p)\\le Z(n)$. The orders $n=7,8,9,12,13,16,17,18,20,21$ are settled by explicit configurations of a different shape, each attaining the cell bound and satisfying $(\\mathrm{RW}3^+)$, obtained from a larger configuration by deleting vertex stars and repairing the restricted grid. At each of these orders $z_2=z_{SL}=z_{RL}=Z(n)$; all remaining orders are conjectural. The first settled order $n=7$ is the only one whose grid has a hole, grounded by a zero-companion rule. Machine-readable configurations and a certificate checker accompany the paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25974",
  "author_total": 3,
  "authors": [
    {
      "name": "Yannan Chen"
    },
    {
      "name": "Johan Löfberg"
    },
    {
      "name": "Liqun Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkza24w8g2qrg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25974",
  "fulltext_url": null,
  "id": "bkza24w8g2qrg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25974",
  "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": "W7214082179",
  "pdf_url": "https://arxiv.org/pdf/2609.25974",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkza24w8g2qrg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.25974",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25974"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families",
  "title_text": "Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/bkza24w8g2qrg",
  "venue": "arXiv",
  "versions": []
}