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Ramsey theory

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A visual argument explaining why the Ramsey number of three is six.
graph theory

A Very Big Small Leap Forward in Graph Theory

By Leila Sloman
May 2, 2023
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Four mathematicians have found a new upper limit to the “Ramsey number,” a crucial property describing unavoidable structure in graphs.

A complex machine produces prime numbers onto a conveyor belt that winds backwards into infinity.
number theory

Why Mathematicians Re-Prove What They Already Know

By Anna Kramer
April 26, 2023
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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.

A panoply of colored fractions
combinatorics

Coloring by Numbers Reveals Arithmetic Patterns in Fractions

By Leila Sloman
March 15, 2023
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In a recent paper, two mathematicians showed that a particular pattern is unavoidable when fractions are categorized.

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.


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