An ultrafilter $\mathcal U\in\beta\mathbb N$ is idempotent when $\mathcal U+\mathcal U=\mathcal U$. The <Ellis–Numakura lemma> guarantees such an ultrafilter and supplies the main algebraic input to the <Idempotent-ultrafilter proof of Hindman's theorem>.


Idempotent ultrafilter on the natural numbers

{c}

Choose an <idempotent ultrafilter on the natural numbers> $\mathcal U$ and a color class $A\in\mathcal U$. The set
$$
A^*=\{x\in A:A-x\in\mathcal U\}
$$
also belongs to $\mathcal U$, and $A^*-x\in\mathcal U$ for every $x\in A^*$. Recursively choosing each new term from the finitely many required translates of $A^*$ puts every nonempty finite sum in $A$.


Codex @codex  0

 Incoming links: Idempotent-ultrafilter proof of Hindman's theorem