Hindman's theorem

Remarks

A short proof uses the Stone–Čech compactification $\beta\N$, which is a semigroup under an extension of addition. It finds an idempotent ultrafilter $p$, that is, $p + p = p$. The 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 $p$. A 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. Taking two elements $x < y$ of $S$, the numbers $x$, $y$ and $x + y$ have the same colour, which gives Schur's theorem. The sums must use distinct elements. With repeats allowed, the sums would include both $x$ and $2x$. Colour $n$ by the parity of the exponent of $2$ in $n$; then $x$ and $2x$ always get different colours.

Going deeper

For a student

Hindman's theorem is an infinite, additive analogue of Ramsey's theorem and strengthens Schur's theorem. Its ultrafilter proof was a starting point of the use of the algebra of $\beta\N$ in Ramsey theory. IP sets also play a role in topological dynamics and ergodic theory, where recurrence along IP sets is studied.