Centenary Mathematical Enigma Solved? The Theorem of Friends and Strangers

Ramsey theory, an enthralling branch of mathematics, is based on a simple yet perplexing principle: within any sufficiently large set, an ordered structure will inevitably emerge. This concept, devised in the 1930s, has spawned a multitude of engaging math challenges, commonly referred to as "Ramsey problems." Among these, the notorious r(4,t) problem has long confounded mathematicians.

Jacques Verstraete and Sam Mattheus, researchers at the University of California, San Diego, have recently unlocked the mystery of r(4,t), a mathematical puzzle that has withstood resolution for decades.


To comprehend the r(4,t) problem, one must first grasp the concept of a graph in graph theory. A graph consists of points connected by lines. Ramsey theory proposes that beyond a certain size, every graph will contain an ordered structure: either a set of points with no lines connecting them (an absence of connection) or a set of points interconnected by all possible lines.

The most familiar example is r(3,3), sometimes termed "the theorem of friends and strangers." It asserts that within a group of six people, there will always be either three people who all know each other or three who are all strangers. The r(3,3) result is six. The apparent simplicity of these problems conceals a remarkable complexity. Suppose the solution for r(5,5) lies between 40 and 50. With 45 points, there would be over 10²³⁴ possible graphs.

Verstraete and Mattheus employed pseudorandom graphs to narrow down estimates of Ramsey numbers. In 2019, they solved r(3,t) using this approach. However, constructing a pseudorandom graph for r(4,t) proved to be a significant challenge.

They delved into additional mathematical areas, including finite geometry, algebra, and probability. Ultimately, their collaboration resulted in a breakthrough: r(4,t) is close to a cubic function of t. This means that for a group where "four individuals all know each other or t individuals are all strangers," it would require roughly t³ people. It's crucial to note that this is an estimation, not an exact answer, but it brings us closer to the truth.

Their finding is currently under review by the Annals of Mathematics, and a preprint is available on arXiv.

This achievement underscores the value of perseverance in mathematics. According to Verstraete, a good problem is resilient and does not yield easily. This resolution is the culmination of many years of dedicated effort, highlighting that even the most daunting problems can be surmounted with determination and creativity.