TITLE: Prove that at a party of $25$ people there is one person knows at least twelve people.
QUESTION [39 upvotes]: So, the full problem goes like this:

There are $25$ people at a party. Assuming that among any three people, at least two of them know each other, prove that there exists one person who must know at least twelve people.

I've been stuck on this problem for a while and haven't really figured out how to proceed. I'm pretty sure that there is an answer that can be found via the pigeonhole principle or some graph theory, but I'm not really sure how to get started. Any help would be appreciated.

REPLY [4 votes]: Proof.
Choose two people $P_1$ and $P_2$ who do not know each other (if you cannot, we are done anyway). Now any third person $P$ either knows $P_1$ or $P_2$, because $\{P,P_1,P_2\}$ forms a group of three people and $\{P_1,P_2\}$ cannot be the pair knowing each other. There are $23$ such other people $P$. So by the pidgeonhole principle one of $P_1$ and $P_2$ must know at least $\lceil 23/2\rceil=12$ of them. $\quad\square$