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. Why can’t there be a graph where lines of different colors remain completely jumbled together?
I put this question to Jonathan Jedwab, a mathematician at Simon Fraser University in British Columbia. He responded with this example, which provides a graphical intuition for why the theorem is true.