coloring

A coloring of a set $X$ by $Y$ is just a function $f:X\to Y$. The term coloring is used because the function can be thought of as assigning a “color” from $Y$ to each element of $X$.

Any coloring provides a partition of $X$: for each $y\in Y$, $f^{-1}(y)$, the set of elements $x$ such that $f(x)=y$, is one element of the partition. Since $f$ is a function, the sets in the partition are disjoint, and since it is a total function, their union is $X$.