# Friendship Theorem

## Contents

## 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 1

This is a simple example of Ramsey's Theorem.

## Proof 2

This can be expressed in graph theoretical language as:

Let the edges of $K_6$, the complete graph of $6$ vertices be edge-colored using $2$ colors.

Then there will be a triangle subgraph of $K_6$ whose edges all have the same color.

Let the colors be **red** and **blue**.

Choose any one vertex $P$.

There are $5$ edges incident to $P$.

They are each colored **red** or **blue**.

By the Pigeonhole Principle, at least $3$ of them must be of the same color.

Without loss of generality, suppose that less than $3$ are **red**.

Then the others, $3$ or more, must be **blue**.

Let $A$, $B$, $C$ be the other vertices incident to these $3$ edges, all of which are **blue**.

There are $2$ cases:

- $(1): \quad$ Let any one of $AB$, $BC$ and $CA$ be
**blue**.

Then that edge, together with the $2$ edges to $P$ forms a **blue** triangle.

- $(2): \quad$ Let none of $AB$, $BC$ and $CA$ be
**blue**.

Then all three edges are **red**.

Thus $AB$, $BC$ and $CA$ form a **red** triangle.

Hence the result.

$\blacksquare$

## Also known as

This theorem has many names and guises; $\mathsf{Pr} \infty \mathsf{fWiki}$ has adopted the name **the friendship theorem** as it is short.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$