Combinatorics

Four mathematicians have found a new upper limit to the “Ramsey number,” a crucial property describing unavoidable structure in graphs.

It’s been known for thousands of years that the primes go on forever, but new proofs give fresh insights into how theorems depend on one another.

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer.

In 50 years of searching, mathematicians found only one example of a “subspace design” in a vector space. A new proof reveals that there are infinitely more out there.

For decades, mathematicians have been inching forward on a problem about which sets contain evenly spaced patterns of three numbers. Last month, two computer scientists blew past all of those results.

In a recent paper, two mathematicians showed that a particular pattern is unavoidable when fractions are categorized.

Mathematicians have uncovered a surprising wealth of rock-paper-scissors-like patterns in randomly chosen dice.

On nights and weekends, Justin Gilmer attacked an old question in pure math using the tools of information theory.

Four Fields Medals were awarded for major breakthroughs in geometry, combinatorics, statistical physics and number theory, even as mathematicians continued to wrestle with how computers are changing the discipline.