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Concept
Theorem on friends and strangers
Summary
The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.
Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:
In any party of six people, at least three of them are (pairwise) mutual strangers or mutual acquaintances.
A proof of the theorem requires nothing but a three-step logic. It is convenient to phrase the problem in graph-theoretic language.
Suppose a graph has 6 vertices and every pair of (distinct) vertices is joined by an edge. Such a graph is called a complete graph (because there cannot be any more edges). A complete graph on vertices is denoted by the symbol .
Now take a . It has 15 edges in all. Let the 6 vertices stand for the 6 people in our party. Let the edges be coloured red or blue depending on whether the two people represented by the vertices connected by the edge are mutual strangers or mutual acquaintances, respectively. The theorem now asserts:
No matter how you colour the 15 edges of a with red and blue, you cannot avoid having either a red triangle—that is, a triangle all of whose three sides are red, representing three pairs of mutual strangers—or a blue triangle, representing three pairs of mutual acquaintances. In other words, whatever colours you use, there will always be at least one monochromatic triangle ( that is, a triangle all of whose edges have the same color ).
Choose any one vertex; call it P. There are five edges leaving P. They are each coloured red or blue. The pigeonhole principle says that at least three of them must be of the same colour; for if there are less than three of one colour, say red, then there are at least three that are blue.
Let A, B, C be the other ends of these three edges, all of the same colour, say blue. If any one of AB, BC, CA is blue, then that edge together with the two edges from P to the edge's endpoints forms a blue triangle.
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