Yes. Associate to each number from 1 to 252 a distinct 5 -element subset of $S=\{1,2, \ldots, 10\}$. Then assign to $(a, b)$ an element of $S$ that is in the subset associated to $a$ but not in that associated to $b$. It's not difficult to see that this numerical assignment is a valid coloring: the color assigned to $(a, b)$ is not in $b$, while the color assigned to $(b, c)$ is in $b$, so they must be distinct.