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combinatorics

Mathematician Lauren Williams standing in front of a whiteboard that has mathematics written on it.
combinatorics

A Mathematician’s Unanticipated Journey Through the Physical World

December 16, 2020

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

Illustration showing a large multicolored graph on top and a smaller one below, which is rising and growing bigger
combinatorics

Disorder Persists in Larger Graphs, New Math Proof Finds

November 4, 2020

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.

Aerial image of a complicated road network.
Abstractions blog

A New Algorithm for Graph Crossings, Hiding in Plain Sight

September 15, 2020

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

Photo of Marijn Heule walking among computer processors
Abstractions blog

Computer Scientists Attempt to Corner the Collatz Conjecture

August 26, 2020

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

Illustration of floating metal cubes joining together, some of their faces yellow
geometry

Computer Search Settles 90-Year-Old Math Problem

August 19, 2020

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.

number theory

Landmark Math Proof Clears Hurdle in Top Erdős Conjecture

August 3, 2020

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

An animation showing a 3-by-3 Latin square, with numbers color coded, and the equivalent graph.
Abstractions blog

‘Rainbows’ Are a Mathematician’s Best Friend

March 18, 2020

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

Animated demonstration of a colorful complete graph being tiled by a smaller tree
combinatorics

Rainbow Proof Shows Graphs Have Uniform Parts

February 19, 2020

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

Photo of a yellow sunflower against a yellow background.
Abstractions blog

Mathematicians Begin to Tame Wild ‘Sunflower’ Problem

October 21, 2019

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.