Frank Ramsey: A more human philosophy

> Ramsey Theory ... is often explained to undergraduates via the Party Problem, which asks what the minimum number of guests is such that at least n will know each other.

This isn't quite correct, as of course you can in principle have a party where no one knows anyone else. The correct question is: "what is the minimum number of guests such that either there are n guests who all know each other, or there are n guests none of whom know each other".

The answer for a given n is sometimes called the nth Ramsey number. Ramsey proved that the Ramsey number exists and is finite for all n. The third Ramsey number is 6, i.e. at a party of 6, there's either a clique of 3 or an "anti-clique" of 3. The fourth Ramsey number is 18 and is the largest n for which we know the number. We know that the fifth Ramsey number lies between 43 and 49 but don't know what it is despite decades of research (a fact that I find amazing).