Recall that in any group of 6 people there are either
3 mutual friends or 3 mutual enemies.
In the group of 10 people consider any person x.
1. x is either friends with 6 people or
enemies with 4 people.
(a) Amongst the 6 friends of x either there are
3 mutual friends or 3 mutual enemies.
Hence including x there are 4 mutual friends or 3 mutual enemies.
(b) Amongst the 4 enemies of x, if two people
are mutual enemies then including x we have
3 mutual enemies. Else the 4 enemies of x are mutual friends.
Hence in either case there are 4 mutual friends or 3 mutual enemies.
2. x is either enemies with 6 people or
friends with 4 people. Note that we have just
exchanged the role of friends and enemies
and so as above we can argue that there are
4 mutual enemies or 3 mutual friends.
Hence we conclude that in any group of 10 people
there are either (4 mutual friends or 3 mutual enemies)
and (3 mutual friends or 4 mutual enemies).
(Answer quoted from
http://www.cse.iitd.ernet.in/~naveen/courses/CSL105/solu5.pdf)