In any meeting of 6 people, does at least one of these two things always happen? - there are 3 people who know each other;
- there are 3 people who are mutual strangers. Explain why.
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Six personas and a triangle social
A meeting of six people seems too small to guarantee such a precise structure. And yet, the problem shows very cleanly how a certain order appears even though no one has planned it.
Hints
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- Pick any person and look at just their five relationships with others.
- Among those five, at least three must be of the same type.
- Now watch what happens between those three people.
Solution
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Answer: Yes, there is always one of the two triangles (known or unknown). Choose any person $X$. He has a relationship with 5 other people. Each relationship is of two types: “knows” or “does not know.” By pigeon, among those 5 relationships at least 3 are of the same type. Case 1: $X$ meets $A,B,C$. - If between $A,B,C$ there is a couple that knows each other (for example $A$ and $B$), then $X,A,B$ form an acquaintance triangle.
- If there are no couples who know each other, then $A,B,C$ are a triangle of strangers. Case 2: $X$ does not know $A,B,C$. It is symmetrical to the previous case: either a triangle of strangers appears with $X$, or a triangle of acquaintances appears within $A,B,C$. In all cases, one of the two triangles exists.
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