We are currently going over pigeonhole principle, ramsey's theorem (graphs and such). Stuck on this particular question:
Within a group of an odd number of people, show that at least one person knows an even number. The person thus is a stranger to an even number of people.
I've been stuck on this one for a while. Please help, thanks!
You must assume that familiarity is a symmetric relation. Otherwise, Alice knows Bob, Bob knows Carl, and Carl knows Alice, will provide a counter-example to your claim. Now, make a graph of who knows who. That is, if Alice knows Bob then Bob knows Alice. Think about counting the number of edges by summing the degrees of each vertex. This will total will be more than the number of edges. That this count must be an integer will give you what you want.