In a party with 2000 persons, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other. How many total people know everyone at the party? Prove your result. (Assume that "knowing" is not necessarily a symmetric relation; that is, if A knows B then B doesn't necessarily know A).
I believe the answer can be anything from $0$ to $1997$. Let $a,b,c$ be the people who don't know anybody and $0$ to $1996$ be everybody else. Clearly all the numbered people know all of $a,b,c$. Now let all the numbered people know everybody except the next number up (including $1996$ doesn't know $0$). Every group of four will include at least one person whose upper neighbor is not there, so will have somebody who knows all the other three. We can restore any of the broken links to raise the number who know everybody.