Probably these two paragraphs sum up the background for the problem nicely

A common analogy for Ramsey theory requires us to consider how many people to invite to a party so that at least three people will either already be acquainted with each other or at least three people will be total strangers to each other.

Here, the Ramsey number, r, is the minimum number of people needed at the party so that either s people know each other or t people don't know each other. This can be written as r(s,t), and we know the answer to r(3,3) = 6.

I was more interested in finding out about Ramsey theorem, though, rather than this new result.