Smallest Number of Strings to Distinguish $n$ Pairwise $L$-distinguishable Strings
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