{
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  "body": "\n## Statement\n\nFor every finite [[graph|00C0BR7D7H]] $H$ there is a constant $\\varepsilon(H) \u003e 0$ with the following property.\nLet $G$ be a graph with $n$ vertices that has no [[induced subgraph|0EREPBVG2G]] isomorphic to $H$.\nThen $G$ has a [[clique|0BYX1D7C2E]] or an [[independent set|08ANQHHG3P]] with at least $n^{\\varepsilon(H)}$ vertices.\n\n## Remarks\n\nWithout the hypothesis on $H$, only a logarithmic bound holds.\nEvery graph on $n$ vertices has a clique or an independent set of size at least $\\tfrac12 \\log_2 n$, by [[Ramsey's theorem|00TQV3F5S4]].\nRandom graphs show that this order $\\log n$ cannot be improved in general.\nErdős and Hajnal proved that excluding $H$ gives a bound of the form $e^{c\\sqrt{\\log n}}$ with $c \u003e 0$ depending on $H$.\nThe conjecture is known for all graphs $H$ with at most four vertices.\nFor $H$ the path on four vertices it is easy: such graphs are perfect, so the product of the clique number and the independence number is at least $n$.\nThe conjecture is open in general; the path on five vertices was settled only in 2023.\n\n## Explanations\n\n### L1 human\n\nRamsey theory says that every large graph has a fairly large set of vertices that are all joined or all unjoined.\nIn general this set may be tiny, about the logarithm of the number of vertices.\nThe conjecture says that forbidding any one fixed pattern as an induced subgraph forces a much larger such set, a power of the number of vertices.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "0EREPBVG2G"
    ],
    "history": {
      "first_stated": "1989",
      "posed_by": "Paul Erdős and András Hajnal"
    },
    "kind": "conjecture",
    "msc": [
      "05C55",
      "05C69"
    ],
    "name": "Erdős–Hajnal conjecture",
    "same_as": {
      "wikidata": "Q16974940",
      "wikipedia": "Erdős–Hajnal conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Hajnal_conjecture"
      }
    ],
    "tag": "PWV2JFX6RR",
    "uses": [
      "00C0BR7D7H",
      "0EREPBVG2G",
      "0BYX1D7C2E",
      "08ANQHHG3P"
    ]
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/PWV2JFX6RR/",
  "links": [
    {
      "field": "about",
      "target": "0EREPBVG2G"
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      "field": "uses",
      "target": "00C0BR7D7H"
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    {
      "field": "uses",
      "target": "0EREPBVG2G"
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      "target": "0BYX1D7C2E"
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      "field": "uses",
      "target": "08ANQHHG3P"
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  "markdown": "https://math2.tamnd.com/node/PWV2JFX6RR.md",
  "msc": [
    "05C55",
    "05C69"
  ],
  "name": "Erdős–Hajnal conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PWV/PWV2JFX6RR.md",
  "status": null,
  "tag": "PWV2JFX6RR",
  "uri": "m2://PWV2JFX6RR",
  "url": "https://math2.tamnd.com/wiki/erdos-hajnal-conjecture/",
  "writing": {
    "tag": "PWV2JFX6RR",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
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        "what": "a source with a locator",
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