The Lawlessness of Large Numbers

So far this year, Quanta Magazine has chronicled three major advances in Ramsey theory, the study of how to avoid creating mathematical patterns. The first result put a new cap on how big a set of integers can be without containing three evenly spaced numbers, like {2, 4, 6} or {21, 31, 41}. The second and third similarly put new bounds on the size of networks without clusters of points that are either all connected, or all isolated from each other.

The proofs address what happens as the numbers involved grow infinitely large. Paradoxically, this can sometimes be easier than dealing with pesky real-world quantities.

For example, consider two questions about a fraction with a really big denominator. You might ask what the decimal expansion of, will “asymptotically” change as   grows. (It gets closer and closer to 0.)