Smallest Number of Strings to Distinguish $n$ Pairwise $L$-distinguishable Strings The following is a homework assignment. I am looking for criticism / feedback on my solution, and I have a specific question.

Suppose $L$ is a language over $\Sigma$, and $x_1, x_2, ... , x_n$ are
strings that are pairwise $L$-distinguishable. How many distinct
strings are necessary in order to distinguish between the $x_i$'s? In
other words, what is the smallest number $k$ such that for some set
$\{z_1, z_2, ...,z_k\}$, any two distinct $x_i$'s are distinguished,
relative to $L$, by some $z_l$? Prove your answer.

The book gives a hint, which reads as follows:

Here is a way of thinking about the question that may make it easier. Think of the $x_i$’s as points on a piece of paper, and think of the $z_l$’s as cans of paint, each $z_l$ representing a different primary color. Saying that $z_l$ distinguishes $x_i$ and $x_j$ means that one of those two points is colored with that primary color and the other isn’t. We allow a single point to have more than one primary color applied to it, and we assume that two distinct combinations of primary colors produce different resulting colors. Then the question is, how many different primary colors are needed in order to color the points so that no two points end up the same color?

I have surely addressed this question Here is my answer.

Solution: Coloring $n$ points so that no two points end up the same color requires $\lceil \log_2 n \rceil$ primary colors.
First enumerate the points using binary representations, and let each digit correspond to a primary color. Then we identify the combination of colors for a given point as all primary colors whose corresponding digit is 1. Each of the points has a unique representation, so each combination of primary colors will be unique.

What do you think of this response? Do you think it needs further explanation?
Also, I am having trouble translating this argument into the context of the original question. How can I better see that the problems are the same?
Your solution is correct but not complete. 


You have not proven the correspondence to the colouring problem given as an intuition in the hint


You could say something on the lines of:- If z distinguishes x and y, we have xz belongs to L but not xy(or vice versa). This is homomorphic(? not sure about this, have to check) to colouring dots x and y with different primary colours, because indistinguishability is transitive, reflexive etc


You have only proven that you have a solution using log n strings but not that log n strings are required


This could be by contradiction. eg. assume that fewer than log n colours can be used to distinguish the dots. Represent the dots as binary strings consisting of the colours they have. Then you have fewer than log n bots required to represent n numbers in binary

IMHO you can skip the colouring and directly prove correspondence with binary strings