The answer to the question that sam512 poses: "How many people must there be in a room, such that necessarily a group of four of them are either mutual friends or mutual strangers?" is 18, and well known.
It can be asked as "What is the smallest possible value of n such that any for any graph G with n or more nodes, G or its complement contains K4" Answering this question is a matter of tedium.
For non-math types, the complement of a graph is the same graph with all the edges removed, and edges put in where there were none before. The complement of Kn is a disconnected set of n nodes
"What is the smallest possible value of N, such that K2^N will force K4 to appear when randomly two-coloured?" is the question that inspired Graham to descibe his famous number, but it is not the same as the question above. The "common man" equivalent can be read below. I find it harder to understand than the graph theory question.
"Consider every possible committee from some number of people n and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find the smallest n that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees."