This question is probably more well known but ah well. I'll try to come up with a less well known one for next day, but this should be a lot of fun for those who haven't seen it:

How many people must attend a party so that there exists a subset of 3 people at the party who all know each other or who all don't know each other? To be clear about definitions, it is understood that if A knows B, then B knows A.

For example, 3 is not enough. As we could have person A knows person B but A does not know C. (and the relationship of B to C doesn't even matter).

Please post a spoiler warning or something if you think/know you have the correct solution...

edit: actually, your wording seems imprecise

Do you mean that there must exist a subset, or that there possibly could exist a subset?

It's a trivial question if you mean that there possibly could exist a subset, so I'm guessing you mean that there must exist a subset no matter what groupings you use.

I think right answer is 3,
at the beginning of the party all three knows all the others. But finally, because they have so great party, they start to talk with bottles and lamps and doesn't recognise each others and then they have some fight and even their mothers doesn't recognise them and when they talk with the toilet seat and want to present themselves they don't remember who they are and just shout GEOOOOOORGE!

Seen it before

I'll hold off

The probability that 3 dudes are either mutual friends or mutual strangers is 100%. In BobbyATA's example, 3 can can be friends, totally strangers (a bit awkward considering that there are not many people who would invite two random guys from the street at his birthday party but not impossible) or John can invite his buddy Joe who comes with his newest girlfriend Sue (neither mutual friends nor mutual strangers). Not 100% guaranteed that the conditions are met.

(Not sure if this was a spoiler or a hint)

sorry Rockman (and others), my wording was imprecise. I meant to say how many party attendees must there be such that there MUST be a subset of 3 guests who...

Also, to Rufus point, yes it would be awkward to have a guest who for example knows nobody, but let us assume this is a possibility (for example lets say perhaps this is an Earth meet and to "know" somebody means to have actually met them before in REAL LIFE, in which case a party guest "not knowing" anyone is quite possible)

An even more fun math question is ; why is it that BobbyATA can not be derivated ?

f(bobby) = bobby^a
f' = a * bobby?

Edit: spoiler, don't read "original post"

fluff^fluff = ffulf^ffulf

Because BobbyATA doest have a function , duh :P

Rufus got it (see his "Original Post") NIce solution man=) Yay math!