{
  "abstract": "We give a new upper bound on the piercing number $τ(\\mathcal{F})$ of a finite family $\\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $τ(\\mathcal{F}) = O(p(\\log\\log p)(\\log\\log\\log p))$, improving the factor $(\\log\\log p)^2$ in the best previously known bound to $(\\log\\log p)(\\log\\log\\log p)$. For each $d>2$, when $\\mathcal{F}$ is a finite family of axis-parallel boxes in $\\mathbb{R}^d$, this yields $τ(\\mathcal{F}) = O(p(\\log p)^{d-2}(\\log\\log p)(\\log\\log\\log p))$. We also study an analogous problem replacing $\\mathbb{R}^d=\\mathbb{R}\\times\\cdots\\times\\mathbb{R}$ with the Cartesian product $T_1\\square\\cdots\\square T_d$ of trees $T_1,\\ldots,T_d$, and boxes with objects of the form $S_1\\square\\cdots\\square S_d$ where $S_i$ is a subtree of $T_i$. For a finite family of such objects under the same intersection assumption as above, we establish an upper bound $τ(\\mathcal{F})=O(p(\\log p)^{d-1})$ for each $d$. We also construct examples showing that this upper bound cannot be improved in order of magnitude.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16703",
  "author_total": 3,
  "authors": [
    {
      "name": "Wei Rao"
    },
    {
      "name": "Taehyun Eom"
    },
    {
      "name": "Minki Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3krgqvch5j90e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16703",
  "fulltext_url": null,
  "id": "3krgqvch5j90e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16703",
  "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",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7221761512",
  "pdf_url": "https://arxiv.org/pdf/2610.16703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3krgqvch5j90e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Piercing boxes and products of trees",
  "title_text": "Piercing boxes and products of trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3krgqvch5j90e",
  "venue": "arXiv",
  "versions": []
}