Mutual friends and strangers

Another puzzle...

There are six people in a group. If you consider any two people in the group, they are either friends to each other, or strangers to each other. Prove that there must exist three people in the group who are mutual friends or mutual strangers.

And now for the challenge puzzle.

There are seventeen people in a much unfriendlier group. If you consider any two people in the group, they are either friends, strangers, or rivals to each other. Prove that there must exist three people in the group who are mutual friends, mutual strangers, or mutual rivals.

(See the solution)