Friendship Theorem/Proof 1

Theorem

Let there be a group of $6$ people.

The traditional setting is that these $6$ people are at a party.

Then (at least) one of the following $2$ statements is true:

$(1): \quad$ At least $3$ of these $6$ people have all met each other before

$(2): \quad$ At least $3$ of these $6$ people have never met each other before.

That is, either there exists a set of $3$ mutual acquaintances, or there exists a set of $3$ mutual strangers.

Proof

This is a simple example of Ramsey's Theorem.

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