There is a group of 100 Readers who come together every month to discuss their findings from the books they have read. They discuss in a group of two people. In order to start a discussion between two people, each individual should have read at least one book that the other person hasn't.
What is the minimum number of distinct books (across all people) that are needed to achieve the condition for any two random members of the group to start a discussion?
As suggested by Almagest, we need $9$ books, with a different subset of $4$ assigned to each reader. There are $\binom{9}{4} = 126$ subsets, so that is a feasible setup for up to $126$ readers. If we had only $8$ books, the largest number of readers we could assign in this way is $\binom{8}{4} = 70$.
That $\binom{9}{4}$ is the largest number of subsets of $9$ objects in which no subset is a subset of another is a consequence of Sperner's Theorem: the largest number of subsets of $n$ objects in which no subset is a subset of another is $$\binom{n}{\lfloor n / 2 \rfloor}$$