CN-TST 2019 Problem 15

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$ , such that there exist a positive integer $k$ , and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?