by Jonathan Kujawa
A few months ago I wrote about one of my favorite results in math: Ramsey's Theorem. It tells us that when we look at things at a large enough scale complete chaos is impossible. That is, if we look hard enough we inevitably find patterns. Call it the Conspiracy Theory Theorem.
Ramsey's theorem launched an entire field of mathematics which answers questions of the form “In such-and-such a setting, what kind of structure do we find if we look on a large enough scale?”. Or you might instead ask: “In such-and-such a setting, if I want to avoid a certain structure on large scales, what do I have to do?”. Of course, it's usually easier to ask the question than to find the answer [1].
A famous recent example is the Green-Tao Theorem. In 2004 Ben Green and Terence Tao proved that within the prime numbers you can find arbitrarily long arithmetic progressions. The prime numbers are the ones which can only be evenly divided by one and themselves (and so have to do with multiplication/division). They are rather randomly distributed amongst all the numbers, but the Green-Tao theorem says that if you look for the right kind of structure (sequences of numbers given by addition) and at a large enough scale, then you can't avoid finding it. It is a striking result which was among the reasons Dr. Tao earned the Fields medal in 2006 and has put Dr. Green in the running for a Fields medal this year [2].
When reading up on Ramsey's Theorem I discovered a delightful book edited by Alexander Soifer entitled “Ramsey Theory: Yesterday, Today, and Tomorrow“. It mixes the history and mathematics of Ramsey theory and covers everything from pre-Ramsey Ramsey theory up to the current state of the art.
From this book I learned of an irresistible 60+ year old question called the Hadwiger-Nelson problem. It's easy to state:
If you want to color the points of the Euclidean plane in such a way as to guarantee that there are never two points of the same color which are exactly one unit apart, how many colors do you need?