I can contribute more to the Alice puzzle:
There are an infinite number of colors, but we know that there can be at most 15 colors at the table, because the Speaker said that every logician at the table can determine their color. That means two people must share every color since otherwise some would not even know that their color existed, and it would be impossible for them to determine their color logically (that is, without guessing). Since 31 divided by 2 is 15 with a remainder, any more than 15 colors would mean that at least one person could never solve the puzzle.
It's reasonable to think that there are fewer than 15 colors because 15 colors would make the problem too easy. Most could look around the table for the one color only represented one time and deduce that they were the other person with the color. After the first round, there would be only three people left, and those three could then deduce that they must all share the same color or one of them could never solve the puzzle.
From that we can determine that at least one color must be represented in a different proportion than the rest. 31 is a prime number, so no matter how many colors are present (15 or fewer) there will always be at least 1 person who cannot be grouped evenly with the rest. As said, that person cannot have their own color (or else they would never be able to logically determine what it was) so necessarily the remainder person must unbalance the group so that one color has a different number of people than any other. This is true even if the colors are not otherwise evenly grouped (for example, if there are 3 blues, 3 greens, 7 reds, 7 yellows, and 4 purples, the purples are represented disproportionately from any other group.)