Theorem on friends and strangers

Statement

Colour each edge of the complete graph $K_6$ red or blue. Then there are three vertices such that the three edges between them have the same colour.

Remarks

Proof: a vertex $v$ has $5$ edges, so at least $3$ of them have one colour, say blue, by the pigeonhole principle. If an edge between their ends is blue, it closes a blue triangle with $v$ . Otherwise these three ends form a red triangle. The number $6$ is the least possible: $K_5$ has a colouring with no triangle of one colour. Thus the Ramsey number $R(3, 3)$ is $6$ .

Reading path

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Complete graph example
Theorem on friends and strangers statement

Tag	`0NYM0ZGAK9`
URI	`m2://0NYM0ZGAK9`
Kind	statement, theorem
Writing	W1 stated
MSC	05C55, 05D10
Status	This copy of the site has no build data. Run `m2 status 0NYM0ZGAK9`.
Uses	Complete graph
Same as	Wikidata Q837506 Wikipedia Theorem on friends and strangers

Theorem on friends and strangers

Statement

Remarks

Reading path

Further reading

What this page needs for W2

What links here