Hint:
Choose a person, P_{1}, to start with. The set of friends of P_{1}
can be denoted F_{1} and the set of non-friends N_{1}. One of these two sets
is infinite. (Possibly both are infinite.)
Let S_{1}=F_{1} if that set is infinite. Otherwise let S_{1} be N_{1}.
Suppose that {P_{1},P_{2},...,P_{n}} and {S_{1},...,S_{n}} have
been defined. Let P_{n+1} be a person in S_{n}. Let S_{n+1} be the set of
people in S_{n} which are friends with P_{n+1} if that set is infinite. Otherwise,
let S_{n+1} be the set of people in S_{n} which are not friends with
P_{n+1}.
In the end you get an infinite sequence of people {P_{i}}, but they are not the set you want. What do you do now?

Complete Solution