I'm confused by this classic graph theory problem:
Suppose there are six people at a party. Prove that it is always possible to find either three people who all know each other, or three people none of whom knows either of the other two.
The wording seems ludicrous! What if one person knows all five of the others, but none of the others knows anyone but that one person who knows all the five? Or any other combination? All six know each other. All six came to a party, none knowing anyone else.
I've looked at other textbooks and they all seem to be similarly worded. It seems the wording should say at least somewhere. For example, if one person knows two to five others, and one of those knows another of that group, then you have the magic edge triangle of three. Likewise, if two to five others are mutual with one, but while you're trying to see if any of those knows another, if the answer is no for three, you get a . . . oh I'm lost! In general it seems to be saying there are either three or more mutual acquaintances, or, if not, than forces the situation to be three or more mutually strangers. What am I missing here?