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combinatorics

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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

By Kevin Hartnett
December 16, 2020
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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

By Kevin Hartnett
November 4, 2020
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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

By Stephanie DeMarco
September 15, 2020
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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

By Kevin Hartnett
August 26, 2020
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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

By Kevin Hartnett
August 19, 2020
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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

By Erica Klarreich
August 3, 2020
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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

By Kevin Hartnett
March 18, 2020
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“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

By Kevin Hartnett
February 19, 2020
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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

By Kevin Hartnett
October 21, 2019
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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.


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