{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $k \\ge 1$ and $n \\ge 1$ be integers, and let $[k] = \\set{1, \\ldots, k}$.\nA *variable word* of length $n$ is a word $w \\in ([k] \\cup \\set{*})^n$ in which the symbol $*$ occurs at least once.\nFor $a \\in [k]$, let $w(a) \\in [k]^n$ be the word obtained from $w$ by replacing every $*$ by $a$.\nThe set $\\set{w(1), \\ldots, w(k)}$ is a *combinatorial line* in $[k]^n$.\n\nLet $k \\ge 1$ and $r \\ge 1$ be integers.\nThen there is an integer $N$ with the following property.\nFor every $n \\ge N$ and every colouring of $[k]^n$ with $r$ colours, some combinatorial line in $[k]^n$ is monochromatic.\n\n## Remarks\n\nExample: for $k = 2$ and $r = 2$ one can take $N = 2$.\nThe square $[2]^2$ has the five lines $\\set{11, 21}$, $\\set{12, 22}$, $\\set{11, 12}$, $\\set{21, 22}$ and $\\set{11, 22}$.\nIn a $2$-colouring, either $11$ and $22$ have the same colour, or the word $12$ has the colour of one of them and lies on a line with it.\nFor $n = 1$ the only line is $\\set{1, 2}$, which is not monochromatic when $1$ and $2$ get different colours.\nThe original proof, by induction on $k$, gives very large bounds; Shelah found a proof with primitive recursive bounds.\nThe theorem implies [[van der Waerden's theorem|V96FRZ2ME1]].\nUse the alphabet $\\set{0, \\ldots, k-1}$ and map a word $w$ to $w_1 + \\cdots + w_n$.\nA combinatorial line goes to an arithmetic progression of length $k$ whose common difference is the number of $*$ symbols.\nThe density version says that, for large $n$, every set containing a fixed positive fraction of $[k]^n$ contains a combinatorial line.\nIt was proved by Furstenberg and Katznelson, and later by an elementary method in the Polymath project.\n\n## Explanations\n\n### L1 human\n\nThink of the game of tic-tac-toe played on a cube with $k$ cells along each side and $n$ dimensions.\nThe theorem says that, if the dimension is large enough, any colouring of the cells with $r$ colours has a full winning line in one colour.\nIn particular, a game of this kind in high enough dimension cannot end in a draw.\n\n### L2 human\n\nThe Hales–Jewett theorem is the abstract core of many Ramsey statements about arithmetic structure.\nIt needs no arithmetic at all: only words and substitution.\nFrom it follow [[van der Waerden's theorem|V96FRZ2ME1]] and its multidimensional form, the Gallai–Witt theorem.\nIts density version similarly implies [[Szemerédi's theorem|B23B0QEMN5]].\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "G5TWXV565S"
    ],
    "kind": "statement",
    "msc": [
      "05D10"
    ],
    "name": "Hales–Jewett theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1032886",
      "wikipedia": "Hales–Jewett theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hales–Jewett_theorem"
      }
    ],
    "tag": "14SGZ5HA4Y"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/14SGZ5HA4Y/",
  "links": [
    {
      "field": "about",
      "target": "G5TWXV565S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/14SGZ5HA4Y.md",
  "msc": [
    "05D10"
  ],
  "name": "Hales–Jewett theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/14S/14SGZ5HA4Y.md",
  "status": null,
  "tag": "14SGZ5HA4Y",
  "uri": "m2://14SGZ5HA4Y",
  "url": "https://math2.tamnd.com/wiki/hales-jewett-theorem/",
  "writing": {
    "tag": "14SGZ5HA4Y",
    "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
      },
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}