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  "body": "\n## Statement\n\nFor a set $S$ of positive integers, let $\\mathrm{FS}(S)$ be the set of all sums $\\sum_{s \\in F} s$ where $F$ is a nonempty finite subset of $S$.\nLet the positive integers be coloured with finitely many colours.\nThen there is an infinite set $S$ of positive integers such that all elements of $\\mathrm{FS}(S)$ have the same colour.\n\n## Remarks\n\nA short proof uses the Stone–Čech compactification $\\beta\\N$, which is a semigroup under an extension of addition.\nIt finds an idempotent ultrafilter $p$, that is, $p + p = p$.\nThe colour class that belongs to $p$ then contains $\\mathrm{FS}(S)$, where the elements of $S$ are chosen one by one inside sets that belong to $p$.\nA set that contains $\\mathrm{FS}(S)$ for some infinite $S$ is called an IP set; the theorem says that in every finite colouring some colour class is an IP set.\nTaking two elements $x \u003c y$ of $S$, the numbers $x$, $y$ and $x + y$ have the same colour, which gives [[Schur's theorem|V6XJQXZ7E8]].\nThe sums must use distinct elements.\nWith repeats allowed, the sums would include both $x$ and $2x$.\nColour $n$ by the parity of the exponent of $2$ in $n$; then $x$ and $2x$ always get different colours.\n\n## Explanations\n\n### L1 human\n\nColour the positive integers with a few colours.\nHindman's theorem says you can find infinitely many numbers such that every sum of finitely many different ones among them has the same colour.\nSo one colour contains a whole infinite family closed under such sums.\n\n### L2 human\n\nHindman's theorem is an infinite, additive analogue of [[Ramsey's theorem|00TQV3F5S4]] and strengthens Schur's theorem.\nIts ultrafilter proof was a starting point of the use of the algebra of $\\beta\\N$ in [[Ramsey theory|G5TWXV565S]].\nIP sets also play a role in topological dynamics and ergodic theory, where recurrence along IP sets is studied.\n",
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