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  "body": "\n## Statement\n\nLet $(P, \\le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] that is not a [[total order|0FRXBVVR03]].\nFor $x, y \\in P$, let $p(x, y)$ be the fraction of the [[linear extensions|W8DZ5KGGHN]] of $P$ in which $x$ comes before $y$.\nThen there are elements $x, y \\in P$ with\n$$\\frac{1}{3} \\le p(x, y) \\le \\frac{2}{3}.$$\n\n## Remarks\n\nThe conjecture is open.\nThe bounds cannot be improved: for $P = \\set{a, b, c}$ with $a \u003c b$ as the only relation, the linear extensions are $cab$, $acb$ and $abc$, so $p(a, c) = 2/3$, $p(c, b) = 2/3$ and $p(a, b) = 1$.\nKahn and Saks proved that a pair with $3/11 \\le p(x, y) \\le 8/11$ always exists.\nBrightwell, Felsner and Trotter improved the constant $3/11$ to $(5 - \\sqrt{5})/10 \\approx 0.276$.\nLinial proved the conjecture for posets of width $2$.\nThe conjecture was later stated again, independently, by Michael Fredman and by Nathan Linial.\n\n## Explanations\n\n### L1 human\n\nSuppose that some comparisons between items are already known, and the goal is to sort them completely.\nThe conjecture says that one can always find a comparison whose answer is not too predictable: each answer occurs in at least one third of the remaining possible orders.\nSuch a comparison cuts the number of possible orders by a fixed factor, which is why the question matters for sorting with partial information.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "W8DZ5KGGHN"
    ],
    "history": {
      "first_stated": "1968",
      "posed_by": "Sergey Kislitsyn"
    },
    "kind": "conjecture",
    "msc": [
      "06A07"
    ],
    "name": "1/3–2/3 conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/1/3–2/3_conjecture"
      }
    ],
    "tag": "AXCH7Z9FKH",
    "uses": [
      "0QV2JS9RJ3",
      "0FRXBVVR03",
      "W8DZ5KGGHN"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/AXCH7Z9FKH/",
  "links": [
    {
      "field": "about",
      "target": "W8DZ5KGGHN"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "0FRXBVVR03"
    },
    {
      "field": "uses",
      "target": "W8DZ5KGGHN"
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  ],
  "markdown": "https://tamnd.github.io/math2/node/AXCH7Z9FKH.md",
  "msc": [
    "06A07"
  ],
  "name": "1/3–2/3 conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AXC/AXCH7Z9FKH.md",
  "status": null,
  "tag": "AXCH7Z9FKH",
  "uri": "m2://AXCH7Z9FKH",
  "url": "https://tamnd.github.io/math2/wiki/1-3-2-3-conjecture/",
  "writing": {
    "tag": "AXCH7Z9FKH",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
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