I know, I know, that parties seem to have become extinct but let us assume that at some point we will again begin to have gatherings of more than just the people in our own households. When that happens, here is a fun exercise you can do. Define as mutual acquaintances as any two people who have met at least once before this occasion, and mutual strangers as any two people who have just met for the first time. It actually does not have to be done at parties but with any group of six or more people.
If the entire group consists of at least some people who do not know every one else, you can take a bet with someone. Let them pick any six people in the group. You bet that that group of six will have a subgroup of three people who are either all mutual acquaintances (i.e., each one has met the other two before) or are mutual strangers (i.e., each person has never met either of the other two people before). You should not actually wager any money on this bet. That would be unfair because you are 100% certain of winning it and you would be taking advantage of the mathematical ignorance of the other person. The result holds true with any group of six people picked at random from anywhere, (say) Facebook or your local community.
Of course, if this is a group of people who all friends and know each other or who are all likely strangers (say people in a café), then the result is obvious. What is not obvious is that this result holds for any group at all that can consist of any combination of mutual acquaintances and mutual strangers. This is because of a theorem from a branch of mathematics known as Ramsey theory. You can see a sketch of the proof of this result here.
Ramsey theory is named after Frank Ramsey who was a mathematics and philosophy prodigy at Cambridge University in the UK who died in 1930 at the very young age of 26, thought to be from a liver infection that he may have picked up by swimming in the river Cam. A review of a biography of Ramsey describes his life and his contributions to mathematics, philosophy, and economics and how despite being so young, he was held in high esteem by already eminent people like the economist John Maynard Keynes and philosophers G. E. Moore and Ludwig Wittgenstein.