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  "body": "\n## Statement\n\nLet $A$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] with rational entries, with columns $c_1, \\dots, c_n \\in \\Q^m$.\nThe system $Ax = 0$ is called partition regular if the following holds for every finite colouring of the positive integers.\nThere is a solution $x = (x_1, \\dots, x_n)$ in positive integers whose entries all have the same colour.\nThe matrix $A$ satisfies the columns condition if the index set $\\set{1, \\dots, n}$ has a partition into nonempty sets $I_1, \\dots, I_k$ with these properties, where $s_j = \\sum_{i \\in I_j} c_i$:\n\n1. $s_1 = 0$.\n2. For each $j$ with $2 \\le j \\le k$, the vector $s_j$ is a linear combination with rational coefficients of the columns $c_i$ with $i \\in I_1 \\cup \\dots \\cup I_{j-1}$.\n\nThen the system $Ax = 0$ is partition regular if and only if $A$ satisfies the columns condition.\n\n## Remarks\n\nFor a single equation $a_1 x_1 + \\dots + a_n x_n = 0$ with nonzero integer coefficients, the columns condition says that some nonempty set of the coefficients $a_i$ has sum $0$.\nSo $x + y - z = 0$ is partition regular, because $1 + (-1) = 0$; this is [[Schur's theorem|V6XJQXZ7E8]].\nThe equation $x + y - 3z = 0$ is not partition regular, because no nonempty subset of $\\set{1, 1, -3}$ sums to $0$.\nThe system $x_i - x_0 - i y = 0$ for $1 \\le i \\le k - 1$, in the unknowns $x_0, \\dots, x_{k-1}, y$, satisfies the columns condition with $I_1 = \\set{x_0, \\dots, x_{k-1}}$ and $I_2 = \\set{y}$.\nIts monochromatic solutions are arithmetic progressions $x_0, x_0 + y, \\dots, x_0 + (k-1)y$ with $y \\ge 1$, so [[van der Waerden's theorem|V96FRZ2ME1]] is a special case.\nThe proof of sufficiency builds the monochromatic solution from many uses of van der Waerden's theorem.\nThe proof of necessity colours each integer by its last nonzero digit in base $p$, for a suitable prime $p$.\n\n## Explanations\n\n### L1 human\n\nColour every positive integer with one of finitely many colours.\nSome systems of linear equations always have a solution inside a single colour, whatever the colouring is.\nRado's theorem tells exactly which systems have this property, with a simple test on the columns of the matrix.\nFor one equation the test is easy: some of the coefficients must add up to zero.\n\n### L2 human\n\nRado's theorem is a central result of [[Ramsey theory|G5TWXV565S]] on the integers.\nIt contains Schur's theorem and van der Waerden's theorem as special cases.\nThe columns condition is checked by linear algebra over $\\Q$, so partition regularity is decidable.\n",
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