Friendship Theorem

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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