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  "body": "\n## Statement\n\nLet $r \\ge 1$ be an integer.\nThen there is a positive integer $N$ with the following property.\nFor every colouring of $\\set{1, \\ldots, N}$ with $r$ colours, there are $x, y, z \\in \\set{1, \\ldots, N}$ of the same colour with $x + y = z$.\nHere $x$ and $y$ need not be distinct.\n\n## Remarks\n\nLet $N + 1$ be at least the Ramsey number for $r$ colours and triangles, given by [[Ramsey's theorem|00TQV3F5S4]].\nColour the edge $\\set{i, j}$ of the complete graph on $\\set{1, \\ldots, N + 1}$ with the colour of $\\abs{i - j}$.\nA monochromatic triangle $i \u003c j \u003c k$ gives $x = j - i$, $y = k - j$ and $z = k - i$ of the same colour with $x + y = z$.\n\nThe least such $N$ is $S(r) + 1$, where the Schur number $S(r)$ is the largest $n$ for which $\\set{1, \\ldots, n}$ can be split into $r$ [[sum-free sets|7SHV2SPYP6]].\nFor example $S(2) = 4$: the colouring $\\set{1, 4}$, $\\set{2, 3}$ has no monochromatic solution, while every $2$-colouring of $\\set{1, \\ldots, 5}$ has one.\nKnown values are $S(1) = 1$, $S(2) = 4$, $S(3) = 13$, $S(4) = 44$ and $S(5) = 160$.\n\n## Explanations\n\n### L1 human\n\nHowever you colour the numbers $1$ to $N$ with a fixed number of colours, if $N$ is large enough, some colour contains two numbers together with their sum.\nFor two colours, $N = 5$ is enough.\nIt is one of the first results of [[Ramsey theory|G5TWXV565S]] in the integers.\n\n### L2 human\n\nSchur proved the theorem in 1916 while studying Fermat's equation modulo primes.\nIt implies that for each $n$ and all large enough primes $p$, the congruence $x^n + y^n \\equiv z^n \\pmod p$ has a solution with $xyz \\not\\equiv 0$.\nThe proof is a direct reduction to [[Ramsey's theorem|00TQV3F5S4]] for triangles.\n[[Rado's theorem|BZSGRCTFH4]] extends it to all linear equations whose monochromatic solutions are forced in this way.\n",
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