introduction to ramsy numbers

Prove that at a party with six people there is a group of at least three mutual acquaintances or three mutual strangers.






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SOLUTION:

Let's call one of the people at the party A. Now, either A has at least three friends present, or, if not, he must be a stranger to at least three people. Now, let's say he has three friends in attendance. If none of the friends know each other, we have a group of three mutual strangers. If two of the people--let's call them B and C--know each other, then we have a group of three mutual friends (that is, A, B and C). A similar argument applies if A has less than three friends (which is equivalent to A being a stranger to at least three people). In this case, if the three strangers all know each other, we have a group of mutual friends. However, if two of these people do not know each other--again, call them B and C--then we clearly have a group of three mutual strangers (namely A,B and C).