Let $\mathbb{N}:=\omega \setminus \{0\}$. Is there $k\in \mathbb{N}$ and a map $c:\mathbb{N}\to \{1,\ldots,k\}$ such that for all $a,b\in\mathbb{N}$ the restriction $c|\_{\{a,\,b,\,a+b\}}$ is non-constant?



Coloring $\mathbb{N}$ such that $\{a, b, a+b\}$ is not monochromatic
No, this is the content of [Schur's Theorem](https://en.wikipedia.org/wiki/Schur%27s_theorem).


