July 24, 2024

Mathematics Problem Solved Decades Later: UC San Diego Mathematicians Make Breakthrough in Ramsey Theory

A team of mathematicians from the University of California (UC) San Diego has finally cracked a puzzling problem in Ramsey theory that has stumped researchers for nearly a century. Jacques Verstraete and Sam Mattheus, the mathematicians responsible for the breakthrough, focused on r(4,t) – a problem that has remained unsolved for over 90 years.

Ramsey theory, a branch of mathematics concerned with the order within structures, was named after the British mathematician and philosopher Frank P. Ramsey. The most well-known problem in this area is r(3,3), also known as the theorem on friends and strangers. It states that in a group of six people, there will always be at least three people who all know each other or three people who all don’t know each other. The answer to r(3,3) is six.

Verstraete explained that this fact holds true regardless of the specific situation or group of people chosen. While it is possible to find more than three people in one clique or the other, there will always be at least three. This seemingly simple problem served as a starting point for mathematicians, who then sought to solve subsequent problems such as r(4,4), r(5,5), and r(4,t) where the number of unconnected points varies.

The answer to r(4,4) was found to be 18 by Paul Erdös and George Szekeres in the last century. However, the solution to r(5,5) remains unknown. Verstraete acknowledged that r(4,t) has been an open problem for over 90 years, but it wasn’t the main focus of his research. Many mathematicians have attempted to solve it without success. Verstraete and Mattheus approached the problem by seeking estimations rather than exact solutions.

Finding these numbers is notoriously difficult, so mathematicians typically look for approximations. Verstraete and Mattheus aimed to determine the best estimates for the Ramsey numbers. Verstraete first became aware of r(4,t) in a book called “Erdös on Graphs: His Legacy of Unsolved Problems” and has kept it in the back of his mind since then. It wasn’t until he collaborated with Mattheus, who specializes in finite geometry, that they began to make real progress.

It took nearly a year, but Verstraete and Mattheus finally found the solution to r(4,t). Their findings revealed that to have a party where there will always be four people who all know each other or t people who all don’t know each other, approximately t3 people need to be present. The mathematicians emphasized that perseverance is key, and one should never give up on solving a problem, no matter how long it takes.

While their research undergoes peer review, Verstraete and Mattheus did not disclose whether they have started working on r(5,5). They credited the difficulty of the mathematical problem as one that makes it intriguing and worth solving. Verstraete humorously noted that he received a call from another mathematician, Fan Chung, who playfully stated that she owed him $250 – a reward initially offered by Erdös in the 1930s for solving the problem.

Although their breakthrough does not include inflation adjustments for the finders’ fee, the mathematicians have indeed made a significant contribution to the field of mathematics with their solution to a problem that has perplexed researchers for almost a century.

1. Source: Coherent Market Insights, Public sources, Desk research
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