{
  "abstract": "Let $\\mathbb{P}(x\\prec y)$ be the probability that $x$ precedes $y$ in a uniformly random linear extension of an $n$-element poset $P$, and define the balancing coefficient to be $δ(x,y)=\\min(\\mathbb{P}(x\\prec y),\\mathbb{P}(y\\prec x))$ with $δ(P)=\\max_{x,y}δ(x,y)$. We prove (Theorem 1) that sufficiently large width forces $δ(P)$ to be arbitrarily close to $1/2$, answering a long-standing conjecture of Kahn and Saks. In fact, we prove the stronger result (Theorem 2) that large width forces one of two configurations in our poset: either a nearly uniform order on $k$ vertices, or an almost fixed order on $t$ vertices with one further vertex inserted uniformly among the $t+1$ slots. We also show that, for fixed $k$, the first possibility must occur within any antichain $X$ of size $Ω(n^{2/3})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30895",
  "author_total": 1,
  "authors": [
    {
      "name": "Max Aires"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/91a8ngnqmxdbj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30895",
  "fulltext_url": null,
  "id": "91a8ngnqmxdbj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30895",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214668768",
  "pdf_url": "https://arxiv.org/pdf/2609.30895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/91a8ngnqmxdbj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proof of the Kahn Saks Conjecture",
  "title_text": "Proof of the Kahn Saks Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/91a8ngnqmxdbj",
  "venue": "arXiv",
  "versions": []
}