Of note, I was operating under a bad assumption with regards to the original example. I assumed that the set was a finite but unknown set of colors or an infinite set of colors. In the former case, every question is giving a little information about the possible set. In the latter it really does not matter much.

A sequence of lilac turquoise turquoise lilac lilac says very much more about what's in the jar than the two hundred color questions omega asked you beforehand.

Yes, this is true. Personally, I am still curious about what to do with the two hundred color questions.