Use the fact that among a group of 10 people, where any two people are either friends or enemies, there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends to show:
Among any group of 20 people, where any two people are either friends or enemies, there are either four mutual friends or four mutual enemies.
I'm unsure on how to approach this problem. How can we use the case of 10 people to generalize the case for 20 people? I've read that if we single out one person, $P_1$, he must have either 10 mutual friends or 10 mutual enemies by the pigeonhole principle. Why is this?
Thank you!