{
  "abstract": "We study the order dimension of Cartesian products of finite posets. Kelly and Trotter conjectured in 1982 that $\\dim(P\\times Q)\\ge\\dim P+\\dim Q-2$ for all finite posets $P$ and $Q$. For $m\\ge3$, let $R_m$ denote the incidence poset of the complete graph on $m$ vertices. We prove that there is a constant $C$ such that, for all sufficiently large $m$, $\\dim(R_m\\times R_m)\\le\\left(1+\\frac{2}{\\log_2 6}\\right)\\dim R_m+C$. Since $1+2/\\log_2 6<2$, this disproves the Kelly--Trotter conjecture and shows that $2\\dim R_m-2-\\dim(R_m\\times R_m)$ can grow linearly with $\\dim R_m$. For every integer $d\\ge8$, we construct an incidence poset $Q_d$ such that $\\dim Q_d=d$ and $Q_d$ has the $(3,d)$-covering property. Consequently, $\\dim(P\\times Q_d)\\le\\dim P+d-3$ for every poset $P$ with $\\dim P\\ge3$. Thus, for every integer $d\\ge8$, the poset $Q_d$ violates the Kelly--Trotter conjecture with every poset of dimension at least $3$. Finally, we prove that a poset $Q$ has an $(r,s)$-covering if and only if $\\dim(Q\\times C^r)\\le s$ for every finite chain $C$ with at least two elements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31441",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhaochen Dong"
    },
    {
      "name": "Kaiyun Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vnw8f94pznmr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31441",
  "fulltext_url": null,
  "id": "9vnw8f94pznmr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31441",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214588458",
  "pdf_url": "https://arxiv.org/pdf/2609.31441",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vnw8f94pznmr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31441",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31441"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Kelly--Trotter conjecture and dimension of poset products",
  "title_text": "The Kelly--Trotter conjecture and dimension of poset products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/9vnw8f94pznmr",
  "venue": "arXiv",
  "versions": []
}