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  "body": "\n## Statement\n\nThe strengthened finite Ramsey theorem is a statement about colorings and [[natural numbers|0MWKJ44V5A]] and states that:\n\nFor any positive integers *n*, *k*, *m*, such that *m ≥ n*, one can find *N* with the following property: if we color each of the *n*-element subsets of *S* = {1, 2, 3,..., *N*} with one of *k* colors, then we can find a subset *Y* of *S* with at least *m* elements, such that all *n*-element subsets of *Y* have the same color, and the number of elements of *Y* is at least the smallest element of *Y*.\n\nWithout the condition that the number of elements of *Y* is at least the smallest element of *Y*, this is a corollary of the [[finite Ramsey theorem|00TQV3F5S4]] in $K_{\\mathcal{P}_n(S)}$, with *N* given by:\n\n$$\\binom{N}{n} = |\\mathcal{P}_n(S)| \\ge R(\\,\\underbrace{ m, m, \\ldots , m }_k\\,).$$\n\nMoreover, the strengthened finite Ramsey theorem can be deduced from the [[infinite Ramsey theorem|00TQV3F5S4]] in almost exactly the same way that the finite Ramsey theorem can be deduced from it, using a compactness argument (see the article on [[Ramsey's theorem|00TQV3F5S4]] for details). This proof can be carried out in second-order arithmetic.\n\nThe Paris–Harrington theorem states that the strengthened finite Ramsey theorem is not provable in Peano arithmetic.\n",
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