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  "body": "\n## Statement\n\nCall a finite set $Y \\subseteq \\N$ *large* if $\\abs{Y} \\ge \\min Y$.\nThe *strengthened finite Ramsey theorem* $\\mathrm{PH}$ is the following statement.\nFor all positive integers $n$, $k$, $m$ there is an integer $N$ with this property.\nFor every colouring of the $n$-element subsets of $\\set{1, 2, \\ldots, N}$ with $k$ colours, there is a large set $Y \\subseteq \\set{1, \\ldots, N}$ with $\\abs{Y} \\ge m$.\nMoreover, all $n$-element subsets of $Y$ have the same colour.\n\n1. $\\mathrm{PH}$ is true; it follows from the infinite Ramsey theorem.\n2. $\\mathrm{PH}$, written as a sentence of first-order arithmetic, is not provable in [[Peano arithmetic|0QFT7Z76N5]] $\\mathsf{PA}$.\n\n## Remarks\n\nPart 1 is a compactness argument.\nIf $\\mathrm{PH}$ failed for some $n, k, m$, the bad colourings for larger and larger $N$ would fit together by König's lemma.\nThey would give a colouring of the $n$-element subsets of $\\N$ whose restriction to each $\\set{1, \\ldots, N}$ is bad.\nThe infinite Ramsey theorem gives an infinite homogeneous set $H$ for it.\nAn initial segment of $H$ with $\\max(m, \\min H)$ elements is then large and homogeneous, a contradiction.\nPart 2 is proved with models.\nIn a non-standard model of $\\mathsf{PA}$, $\\mathrm{PH}$ gives initial segments that are themselves models of $\\mathsf{PA}$.\nFrom this, $\\mathsf{PA}$ plus $\\mathrm{PH}$ proves the consistency of $\\mathsf{PA}$.\nThe function that sends $(n, k, m)$ to the least $N$ grows faster than every function that $\\mathsf{PA}$ proves total.\n\nWithout the word \"large\", the statement is the ordinary finite Ramsey theorem, which is provable in $\\mathsf{PA}$.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a natural statement about colouring finite sets that is true but cannot be proved from the usual axioms of arithmetic.\nIt differs from the finite Ramsey theorem by one small demand: the same-coloured set must have at least as many elements as its smallest member.\nThis small change makes the required numbers grow so fast that arithmetic cannot prove they exist.\n\n### L2 human\n\nParis and Harrington proved the theorem in 1977.\nIt was one of the first natural examples from combinatorics of a true statement independent of $\\mathsf{PA}$, after the artificial sentences of [[Gödel's incompleteness theorems|X9WYNYE29C]].\n[[Goodstein's theorem|XQMNJF241J]] is another example of the same kind.\n",
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