Combinatorics

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.

Two computer scientists found — in the unlikeliest of places — just the idea they needed to make a big leap in graph theory.

A powerful technique called SAT solving could work on the notorious Collatz conjecture. But it’s a long shot.

By translating Keller’s conjecture into a computer-friendly search for a type of graph, researchers have finally resolved a problem about covering spaces with tiles.

Two mathematicians have proved the first leg of Paul Erdős’ all-time favorite problem about number patterns.

“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.

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.

Polynomials aren’t just exercises in abstraction. They’re good at illuminating structure in surprising places.