Combinatorics

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

Fifty years ago, Paul Erdős and two other mathematicians came up with a graph theory problem that they thought they might solve on the spot. A team of mathematicians has finally settled it.

The mathematician Federico Ardila takes a creative approach to the search for useful answers hiding among inconceivably huge numbers of possible ones.

Avi Wigderson and László Lovász won for their work developing complexity theory and graph theory, respectively, and for connecting the two fields.

Po-Shen Loh has harnessed his competitive impulses and iconoclastic tendencies to reinvigorate the U.S. Math Olympiad program.

Lauren Williams has charted an adventurous mathematical career out of the pieces of a fundamental object called the positive Grassmannian.

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.