Ramsey theory

A new proof significantly strengthens a decades-old result about the ubiquity of ways to represent whole numbers as sums of unit fractions.

A new paper shows how to create longer disordered strings than mathematicians had thought possible, proving that a well-known recent conjecture is “spectacularly wrong.”

Researchers have proved a special case of the Erdős-Hajnal conjecture, which shows what happens in graphs that exclude anything resembling a pentagon.

At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.

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.