{
  "abstract": "We study two questions about the second-order Zarankiewicz numbers. The first is the recursive-line equality $z_2(m,n)=z_{RL}(m,n)$, known to hold in every computed case and conjectured in general. The second, here called \\emph{overtake}, is the strict inequality $z_{2A}(m,n)>z_2(m,n)$ between the second-order augmented number and the second-order number. Both are governed by the number $H_{\\mathrm{limited}}(m,n)$ of unoccupied cells in a limited extremal configuration. We prove that the first increase $z_2\\to z_2+1$ already costs three unoccupied cells, forcing a unit of one-edge deficit, and the cell identity then bounds the excess $z_{2A}-z_2$ by $\\lfloor(H_{\\mathrm{limited}}-1)/2\\rfloor$. In particular, when the recursive-line hole count is at most two, no overtake occurs. We conjecture the sharp budget $\\lfloor H_{\\mathrm{limited}}/3\\rfloor$ and that $H_{\\mathrm{limited}}\\ge5$ is the threshold for an overtake. In the computed range $n\\le7$, no overtake occurs at any order, the sole exception being the known case $(4,4)$. The recursive-line equality $z_2=z_{RL}$ is verified at every order in the range where both values are determined; the orders at which it remains unsettled are $(12,4)$ and $(14,4)$ at width four, $(11,5)$ at width five, and $(8,7)$ at width seven.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22875",
  "author_total": 2,
  "authors": [
    {
      "name": "Chen Ouyang"
    },
    {
      "name": "Liqun Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jpt35947pt91.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22875",
  "fulltext_url": null,
  "id": "8jpt35947pt91",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22875",
  "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": "W7222952521",
  "pdf_url": "https://arxiv.org/pdf/2610.22875",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jpt35947pt91.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22875",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22875"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Overtake and the recursive-line equality for second-order Zarankiewicz numbers",
  "title_text": "Overtake and the recursive-line equality for second-order Zarankiewicz numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/8jpt35947pt91",
  "venue": "arXiv",
  "versions": []
}