Ramsey theory

David Conlon and Asaf Ferber have raised the lower bound for multicolor “Ramsey numbers,” which quantify how big graphs can get before patterns inevitably emerge.

“Rainbow colorings” recently led to a new proof. It’s not the first time they’ve come in handy.

Mathematicians have proved that copies of smaller graphs can always be used to perfectly cover larger ones.

We finally know how big a set of numbers can get before it has to contain a pattern known as a “polynomial progression.”

Mathematicians have figured out exactly how many moves it takes to randomize a 15 puzzle.

A major advance toward solving the 60-year-old sunflower conjecture is shedding light on how order begins to appear as random systems grow in size.

Paul Erdős placed small bounties on hundreds of unsolved math problems. Over the past 20 years, only a handful have been claimed.

The three young friends who devised the “happy ending” problem would become some of the most influential mathematicians of the 20th century, but were never able to solve their own puzzle. Now it receives its first big breakthrough.

Ramsey’s theorem predicts a surprising (and useful) consistency in the organization of graphs. Here’s a simple visual proof of how it works.