Researchers at the University of California San Diego, Jacques Verstraete and Sam Mattheus, have recently made a breakthrough in solving a longstanding Ramsey problem that had puzzled mathematicians for decades. Ramsey theory deals with finding order within large graphs, and the problem in question was finding the answer to r(4, t), where t represents points without lines connecting them. The solution to r(3, 3), a well-known Ramsey problem, implies 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.
After solving r(3, 3), mathematicians sought to unravel r(4, 4), r(5, 5), and r(4, t), where t varies. The solution to r(4, 4) was found to be 18, thanks to a theorem developed in the 1930s by Paul Erdös and George Szekeres. However, the solution to r(5, 5) remains unknown, showcasing the difficulty in solving these problems. It can be a daunting task to estimate the answers to these problems, as the sheer number of graphs to consider can be overwhelming.
Verstraete had been aware of the r(4, t) problem for most of his professional career and had encountered it in Erdös on Graphs: His Legacy of Unsolved Problems. The problem, initially posed by Erdös, offered a reward to anyone who could solve it. Despite being an open problem for over 90 years, progress had been slow due to the complexity involved. Verstraete and Mattheus eventually cracked the problem by incorporating pseudorandom graphs from finite geometry, achieving an estimate that r(4, t) is close to a cubic function of t.
Their findings, currently under review with the Annals of Mathematics, reveal that after years of hard work and collaboration, they were able to make significant progress in solving a challenging mathematical problem. Verstraete emphasizes the importance of perseverance in tackling difficult problems, as the process of finding a solution can be lengthy and require innovative approaches. The researchers’ success in solving r(4, t) serves as a testament to the value of persistence and determination in the field of mathematics.
The discovery of using pseudorandom graphs from finite geometry to assist in solving Ramsey problems represents a breakthrough in the field, as it provides a new approach to estimating solutions for these notoriously difficult problems. By combining expertise from various mathematical disciplines and dedicating years to unraveling the mystery of r(4, t), Verstraete and Mattheus have demonstrated the power of collaboration and creative problem-solving in mathematics. Their achievement not only sheds light on the elusive solutions to Ramsey problems but also inspires future mathematicians to tackle challenging problems with perseverance and resilience.
