Why Must Six People Include Three Mutual Acquaintances or Three Mutual Strangers?

Suppose six people attend a gathering and every pair has only one of two relationships: they know each other, or they are strangers. No matter how the fifteen pairwise relationships are arranged, there must be either three people who all know one another or three who are all mutual strangers. This is not merely likely. It is a certainty in combinatorics, written as R(3,3)=6.

Choose one person, A. Each of A’s five relationships has one of two types, so at least three of the other people must have the same type of relationship with A. Suppose B, C and D all know A. Now inspect the relationships among B, C and D. If any pair know each other, that pair together with A forms a trio of mutual acquaintances. If none of the three pairs know each other, then B, C and D form a trio of mutual strangers. The argument is perfectly symmetric if the initial three are all strangers to A.

The striking point is that mathematics does not predict who knows whom. It proves that once enough local relationships are present, every neat three-person pattern cannot be avoided. Five people are not enough: place them around a pentagon, make neighbours acquaintances and non-neighbours strangers, and both types of trio can be avoided. Six is therefore the smallest number that guarantees the result.

In real life, acquaintance has degrees and may not always be symmetrical. The theorem simplifies it to a binary, symmetric relation. Its force comes from that precise model, so it should not be transferred unchanged to ambiguous social life.

https://mathworld.wolfram.com/RamseyNumber.html