Answer for Challenging Question#2

Let's try to construct a party that does not have three mutual friends or three mutual strangers. Suppose you are one of the six people at the party. Let's say you know three other people. We'll call them Alice, Bob, and Carol. If Alice and Bob also know each other, then we have three people, you Alice and Bob, who know each other. We're trying to construct a party where this doesn't happen, so Alice and Bob don't know each other. The same holds for Alice and Carol, and Bob and Carol. Thus the three people that you know, don't know each other, giving a triangle of strangers.
Now if you only know two people at the party, or one person, or perhaps nobody, then there are three people at the party who are strangers to you. Once again we'll call them Alice, Bob, and Carol. If Alice and Bob don't know each other, we have a triangle of strangers. So Alice and Bob do know each other, along with Alice and Carol, and Bob and Carol. These three form a triangle of friends. No matter how we arrange things, there will be three guests who know each other or three guests who are mutual strangers.