{
  "abstract": "Let $z(n)$ be the maximum number of edges in a tripartite $3$-uniform hypergraph with $n$ vertices in each part and no copy of $K_{2,2,2}^{(3)}$ (a ``box''). Erdős (1964) proved that $z(n) = O(n^{11/4})$, whereas the best previous lower bound, due to Katz, Krop, and Maggioni (2002), was $Ω(n^{8/3})$. For each $q = 2^m$, we construct a box-free hypergraph with $q^4$ vertices in each part and $q^{11}$ edges, showing that $z(n) = Θ(n^{11/4})$. The construction uses the power map $τ(s) = s^{q^2-q+1}$ on $\\F_{q^3}$, which sends the fibers of $s \\mapsto τ(s+1) + τ(s)$ to pairwise skew affine lines over $\\F_q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22381",
  "author_total": 1,
  "authors": [
    {
      "name": "Dean Menezes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0dcbb748m6f1x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22381",
  "fulltext_url": null,
  "id": "0dcbb748m6f1x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22381",
  "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": "W7214019362",
  "pdf_url": "https://arxiv.org/pdf/2609.22381",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0dcbb748m6f1x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.22381",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22381"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Three-Dimensional Erdős Box Problem Has Exponent $11/4$",
  "title_text": "The Three-Dimensional Erdős Box Problem Has Exponent 11/4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/0dcbb748m6f1x",
  "venue": "arXiv",
  "versions": []
}