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)

Solution to measuring problems

These are the solutions to "Two more measuring problems".

Susan of Intrinsically Knotted solved the first one:

Start both ropes burning, the first rope just on one end, and the second rope burning from both ends. The second rope will burn completely in exactly 30 minutes (since it is effectively burning twice as fast), at the end of which you will have exactly 30 minutes of rope left on the first rope. Now you can burn the first rope from both ends and it will take exactly 15 minutes.

Eduard's friend Constantino solved the second one in eight steps. This is impressive, since I was only previously aware of a nine-step solution.

Notation is X/Y where X is number of cups of water, and Y is the number of cups of lemonade. The containers are listed in order from smallest to largest.

Start: 0/0, 0/10, 0/0

1. 0/6, 0/4, 0/0
2. 0/0, 0/4, 0/6
3. 6/0, 0/4, 0/6
4. 0/0, 0/4, 6/6
5. 0/4, 0/0, 6/6
6. 1/5, 0/0, 5/5
7. 0/0, 1/5, 5/5
8. 0/0, 5/5, 5/5

And there are lots of other solutions you can think up, I'm sure.