Hi Everyone!
Just wondering, in mathematical terms, how would you prove that R(3,3) = 6?
Also, if R(3,3) = 6 and R(2,4) = 4, how would you prove that R(3,4) ≤ 10?
Cheers in advance!
Banana
What your question really is asking is the following. You have 6 people at a party, given any two people you can conclude that they did one of the following: slept together or ignored eachother. Pick any person in the party, call him A. Between A and B, A and C, A and D, A and E, A and F we have exactly one of these two choices: two of them slept together or two of them ignored eachother. By the pigeonhole principle it means that there exists three people, call them B and C and D which did the same thing together with A, say they have both slept with A. If any one of these three people (B,C, or D) slept with another one (B,C or D) then between those two that slept and A we have found a 3 pair of people who have all slept with eachother! Thus, we have a three-group of people who all slept together*. But, if neither these three people (B,C, and D) sleep with eachother than we have a a three-group of people (B,C, and D) that did not sleep with eachother. Thus, we can conclude that in either case we can either find a three-group that slept together or a three-group that did not sleep together.
*)By the way, if three people sleep together then we can conclude that there was some homosexuality going on by a simple application of the pigeonhole principle.