{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nFor [[cardinals|0QKPNZKNEA]] $\\kappa$, $\\lambda$, $\\mu$ and an integer $n \\ge 1$, write $\\kappa \\to (\\lambda)^n_\\mu$ for the following property.\nLet $X$ be a set of size $\\kappa$, and let $c$ map the $n$-element subsets of $X$ to $\\mu$.\nThen some $H \\subseteq X$ of size $\\lambda$ has all its $n$-element subsets of the same colour under $c$.\nFor an infinite cardinal $\\kappa$, let $\\beth_0(\\kappa) = \\kappa$ and $\\beth_{m+1}(\\kappa) = 2^{\\beth_m(\\kappa)}$, as for the [[beth numbers|ZC4BHVHWE9]].\nThen, for each infinite cardinal $\\kappa$ and each integer $n \\ge 0$,\n$$\\beth_n(\\kappa)^+ \\to (\\kappa^+)^{n+1}_\\kappa.$$\n\n## Remarks\n\nThe case $n = 0$ is the pigeonhole principle: a map from $\\kappa^+$ points to $\\kappa$ colours is constant on $\\kappa^+$ points.\nThe case $n = 1$ reads $(2^\\kappa)^+ \\to (\\kappa^+)^2_\\kappa$, and for $\\kappa = \\aleph_0$ it gives $(2^{\\aleph_0})^+ \\to (\\aleph_1)^2_{\\aleph_0}$.\nThe proof goes by induction on $n$.\nIt builds a long sequence in which the colour of an $(n+1)$-element set depends only on its $n$ smallest elements, and then applies the case $n - 1$.\nThe bound is sharp for $n = 1$: by a theorem of Sierpiński, $2^{\\aleph_0} \\not\\to (\\aleph_1)^2_2$, so $2^{\\aleph_0}$ points do not suffice even for two colours.\nThe theorem extends the infinite [[Ramsey theorem|00TQV3F5S4]] $\\aleph_0 \\to (\\aleph_0)^n_k$ to uncountable homogeneous sets.\n\n## Explanations\n\n### L1 human\n\nColour the pairs of points of a set with a small number of colours.\nRamsey's theorem says that an infinite set always contains an infinite set whose pairs all have one colour.\nFor uncountable sets this is no longer free: to get an uncountable one-coloured set, the starting set must be large.\nThe Erdős–Rado theorem says how large is enough: more than continuum many points always suffice for pairs.\n\n### L2 human\n\nThe notation $\\kappa \\to (\\lambda)^n_\\mu$ is the arrow notation of the partition calculus of Erdős and Rado.\nThe theorem is the basic positive result of uncountable Ramsey theory, and Sierpiński's colouring shows that it cannot be improved for pairs.\nIt is used in model theory, for example to find indiscernible sequences in large models, and in the study of large cardinals defined by partition properties.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "0QKPNZKNEA"
    ],
    "kind": "statement",
    "msc": [
      "03E02",
      "05D10"
    ],
    "name": "Erdős–Rado theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1366581",
      "wikipedia": "Erdős–Rado theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Erdős–Rado_theorem"
      }
    ],
    "tag": "S1BHE358VX",
    "uses": [
      "0QKPNZKNEA",
      "ZC4BHVHWE9",
      "00TQV3F5S4"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/S1BHE358VX/",
  "links": [
    {
      "field": "about",
      "target": "0QKPNZKNEA"
    },
    {
      "field": "uses",
      "target": "0QKPNZKNEA"
    },
    {
      "field": "uses",
      "target": "ZC4BHVHWE9"
    },
    {
      "field": "uses",
      "target": "00TQV3F5S4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/S1BHE358VX.md",
  "msc": [
    "03E02",
    "05D10"
  ],
  "name": "Erdős–Rado theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S1B/S1BHE358VX.md",
  "status": null,
  "tag": "S1BHE358VX",
  "uri": "m2://S1BHE358VX",
  "url": "https://math2.tamnd.com/wiki/erdos-rado-theorem/",
  "writing": {
    "tag": "S1BHE358VX",
    "kind": "statement",
    "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
      }
    ]
  }
}