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Ramsey theory

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A rectangle broken up into five fractions.
number theory

Math’s ‘Oldest Problem Ever’ Gets a New Answer

By Jordana Cepelewicz
March 9, 2022
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A new proof significantly strengthens a decades-old result about the ubiquity of ways to represent whole numbers as sums of unit fractions.

An illustration of a mathematician on a ladder stringing red and blue beads against a tiled red backdrop dotted with blue circles.
combinatorics

Mathematician Hurls Structure and Disorder Into Century-Old Problem

By Erica Klarreich
December 15, 2021
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A new paper shows how to create longer disordered strings than mathematicians had thought possible, proving that a well-known recent conjecture is “spectacularly wrong.”

graph theory

New Proof Reveals That Graphs With No Pentagons Are Fundamentally Different

By Steve Nadis
April 26, 2021
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Researchers have proved a special case of the Erdős-Hajnal conjecture, which shows what happens in graphs that exclude anything resembling a pentagon.

Photo of Ashwin Sah standing in front of a sculpture in a park
Abstractions blog

Undergraduate Math Student Pushes Frontier of Graph Theory

By Kevin Hartnett
November 30, 2020
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At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.

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.

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.

An illustration of a woman sitting in a field embroidering a flower pattern. Around her grow wildflowers that appear to be randomly distributed but whose colors reveal a hidden pattern.
number theory

Mathematicians Catch a Pattern by Figuring Out How to Avoid It

By Kevin Hartnett
November 25, 2019
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We finally know how big a set of numbers can get before it has to contain a pattern known as a “polynomial progression.”

Abstractions blog

Mathematicians Calculate How Randomness Creeps In

By Marcus Woo
November 12, 2019
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Mathematicians have figured out exactly how many moves it takes to randomize a 15 puzzle.


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