Monday 03 March 2025
The Erdos-Szekeres Problem has been a thorn in the side of mathematicians for nearly a century, and recently, researchers have made significant progress in cracking this stubborn nut. In essence, the problem asks how many points are needed to guarantee that there exists either a set of points collinear (lying on the same line) or a convex polygon with a specific number of sides.
The problem’s origins date back to 1935, when Hungarian mathematicians Paul Erdos and George Szekeres stumbled upon it while exploring combinatorial geometry. Since then, numerous attempts have been made to solve it, but progress has been slow going. That is, until recently.
A team of researchers from Kyoto University in Japan has made a significant breakthrough, providing an upper bound for the problem that’s exponentially tighter than previous estimates. To put this into perspective, their new calculation shows that with just 2n+O(√n log n) points in three-dimensional space, it’s virtually guaranteed that either there will be three collinear points or a convex polygon with n sides.
So, what does this mean exactly? Well, consider the following thought experiment. Imagine you have a set of random points scattered throughout three-dimensional space. You might expect that it would take an enormous number of points before you’d stumble upon a collinear trio or a convex polygon with a specific number of sides. The Kyoto team’s results show that this intuition is correct – but only up to a point.
In reality, the number of points needed to guarantee such formations is much lower than previously thought. This has significant implications for fields like computer science and engineering, where optimizing spatial arrangements of objects is crucial. For instance, in robotics, understanding how to arrange sensors and actuators in three-dimensional space can be critical for achieving precise movements.
The Kyoto team’s breakthrough is also shedding light on a related problem known as the Ramsey-remainder problem. In essence, this asks what happens when you remove certain points from a set, but still require that either collinear or convex polygon formations exist. The researchers’ findings have far-reaching implications here too, potentially leading to new insights into complex systems and networks.
What’s remarkable about this work is not just the sheer mathematical prowess required to tackle it, but also the interdisciplinary nature of the research. Mathematicians, computer scientists, and engineers are all converging on this problem from different angles, each bringing their unique expertise to bear.
Cite this article: “Cracking the Erdos-Szekeres Problem: A Breakthrough in Combinatorial Geometry”, The Science Archive, 2025.
Erdos-Szekeres Problem, Combinatorial Geometry, Kyoto University, Upper Bound, Convex Polygon, Collinear Points, Three-Dimensional Space, Computer Science, Engineering, Ramsey-Remainder Problem
Reference: Koki Furukawa, “Convex polytopes with coplanarity” (2025).
