Answer
--Showing that in a group of 10 people , there are either three mutual
friends or four mutual enemies, and there are either
three mutual enemies or four mutual friends.
Work Step by Step
--By symmetry we need prove only the first statement.
- Let A be one of the people.
-Either A has at least four friends,
-or A has at least six enemies among the other nine people (because
3 + 5 < 9).
-Suppose, in the first case, that B, C, D, and E are all A’s friends. If any two of these are friends with each other, then we have found three mutual friends.
-Otherwise {B, C,D, E} is a set of four mutual enemies. In the second
-case, let {B, C, D, E, F, G} be a set of enemies of A.
- , among B, C, D, E, F, and G there are either three mutual friends or three mutual enemies, who form, with A, a set of four mutual enemies.
