{
  "abstract": "We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\\times2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let ${z_A}(m,n)$ be the optimum over all $C_4$-free initial matrices, and ${z_L}(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\\to\\infty$ with $n\\le m=o(n^2)$, we prove \\[ {z_A}(m,n)-{z_L}(m,n)\\ge\\left(\\frac1{30}-o(1)\\right)mn \\] and determine the sharp second-order term: \\[ {z_A}(m,n)=\\frac{mn}{3}+\\left(\\frac1{\\sqrt6}+o(1)\\right)n\\sqrt m. \\] An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\\to\\infty$ and $m/n^2\\to c>0$, the limited density ${z_L}(m,n)/(mn)$ tends to $1/3$ if and only if $c\\ge1/12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16555",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikita Lebedev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ec0rwcjj79wg6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16555",
  "fulltext_url": null,
  "id": "ec0rwcjj79wg6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16555",
  "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": "W7213396083",
  "pdf_url": "https://arxiv.org/pdf/2609.16555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ec0rwcjj79wg6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Density and separation for augmented Zarankiewicz numbers",
  "title_text": "Density and separation for augmented Zarankiewicz numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/ec0rwcjj79wg6",
  "venue": "arXiv",
  "versions": []
}