[EventHorizon]

There are n people in a room. Between each pair of people, they either hate each other or like each other (e.g., the feeling is always mutual).

(1) What is the minimum number of people, n, such that you are guaranteed to have either a group of three that all like each other or a group of three that all hate each other?

(2) Now we add in the ability to be indifferent (not like or hate). What must n be such that there is guaranteed to be a group of three that all like each other, all hate each other, or are all indifferent towards each other?

(3) Now say there are m different emotions. How many people must there be such that there is guaranteed to be a group of x people that all have the same emotion towards each other?

Edited February 13, 2009 by EventHorizon

[EventHorizon]

Hey tarunark,

You got question 1, but missed 2 and 3. Three, as originally written, may be unsolvable....and definitely too hard and/or time consuming for most braindenners... let alone career mathematicians (it would require significant research and/or mathematical knowledge (it is an unsolved problem in Ramsey Theory)).

Good job on question 1 though.