Pigeonhole principle applied to "friendship" The website BaceFook permits users to post pictures, stories and to share them with their “friends”. Any two users are either “friends” or not “friends”. Note that if A is a “friend” of B then B is also a “friend” of A since “friend”ship is reciprocal. Everyone is always their own “friend” since it is sad to have no “friends” There are n users on BaceFook. Prove that at any moment, two of these have the same number of “friends” (not necessarily the same “friends” but the same number).
I know this is a pigeonhole problem, but i really need some help with it.

With the condition that $n>1$:
If there is someone (call them Xena) with the maximum possible $n$ friends, the minimum number of friends anyone else can have is $2$, which would be Xena and themselves. So the range of possible number of friends is $[2,n]$.
If there is not anyone with $n$ friends, the range of possible number of friends is $[1,n-1]$.
Either way there are $n$ people and less than $n$ possibilities for friend counts, so there must be people with identical friend counts by the pigeonhole principle.
Note that the logic also works if you do not count as your own friend; the ranges just shift down by one at both limits.
