A Simple Visual Proof of a Powerful Idea in Graph Theory - Facts So Romantic

A recent advance in geometry makes heavy use of Ramsey's theorem, an important idea in another field--graph theory. Ramsey's theorem states that in any graph where all points are connected by either red lines or blue lines, you're guaranteed to have a large subset of the graph that is completely uniform--that is, either all red or all blue. Equivalently, you can go the other way: Pick how big you want your uniform subset to be. Ramsey's theorem states that somewhere out there there's a graph in which a subset of that size must arise. It's not obvious why this is true.