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  "body": "\n## Statement\n\nLet $r \\ge 1$ and $k \\ge 1$ be integers.\nThen there is a positive integer $N$ with the following property.\nLet $\\set{1, \\ldots, N}$ be coloured with $r$ colours.\nThen there are integers $a \\ge 1$ and $d \\ge 1$ such that the $k$ numbers $a, a + d, \\ldots, a + (k-1)d$ lie in $\\set{1, \\ldots, N}$ and all have the same colour.\n\n## Remarks\n\nThe least such $N$ is the van der Waerden number $W(r, k)$.\nFor example $W(2, 3) = 9$: the colouring $1, 2, 5, 6$ red and $3, 4, 7, 8$ blue of $\\set{1, \\ldots, 8}$ has no monochromatic three-term progression, while every $2$-colouring of $\\set{1, \\ldots, 9}$ has one.\nThe classical proof is a double induction with \"colour focusing\" and gives bounds that grow like the Ackermann function.\nThe theorem also follows from the [[Hales–Jewett theorem|14SGZ5HA4Y]]: a combinatorial line in $\\set{0, \\ldots, k-1}^n$ is sent to an arithmetic progression by the map $w \\mapsto w_1 + \\cdots + w_n$.\nShelah gave primitive recursive bounds, and Gowers gave the bound $W(r, k) \\le 2^{2^{r^{2^{2^{k + 9}}}}}$ through his proof of [[Szemerédi's theorem|B23B0QEMN5]].\n\n## Explanations\n\n### L1 human\n\nHowever you colour the positive integers with finitely many colours, one colour contains arithmetic progressions of every length.\nThe theorem says more: for a given length and number of colours, a fixed finite stretch $1, \\ldots, N$ already suffices.\n\n### L2 human\n\nVan der Waerden's theorem is a cornerstone of [[Ramsey theory|G5TWXV565S]] in the integers.\nIts density form is [[Szemerédi's theorem|B23B0QEMN5]]: a set of integers of positive upper density contains arbitrarily long progressions.\nThe colour classes of a colouring cannot all have density zero, so the density form implies the colouring form.\nThe [[Green–Tao theorem|H8BAHWSEMC]] gives progressions in the primes, which have density zero.\nExact values of $W(r, k)$ are known only for a few small cases.\n",
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