Young mathematicians have achieved a significant breakthrough in combinatorics by solving a long-standing problem in number theory. Ashvin SAH and Metab Suni, who started their scientific endeavors at the Massachusetts Institute of Technology (MIT), collaborated with James Leng, a graduate student at the University of California, Los Angeles (UCLA), to develop a new method for assessing sets of integers that do not contain arithmetic progressions. Together, they produced an impressive 57 mathematical proofs, many of which have been recognized as significant achievements in various fields.
Arithmetic progressions are sequences of numbers with equal intervals, such as {9, 19, 29, 39, 49}. Despite their apparent simplicity, these sequences possess a complex mathematical structure that presents a significant challenge to study.
The concept of sets containing no arithmetic progressions was first proposed in 1936 by mathematicians Erdesh and Pal Turan. They posited that any set containing a significant portion of integers must necessarily include arbitrarily long arithmetic progressions. In 1975, mathematician Andras Szemeredi proved this hypothesis, paving the way for new avenues of mathematical research.
In the late 1990s, Timothy Gowers, a mathematician currently working at the College de France, developed a theory that addressed this challenge. Subsequently, in 2001, he applied his methods to Szemeredi’s theorem, establishing the optimal size limit for sets that avoid arithmetic progressions of any length. While mathematicians utilized Gowers’ framework to solve other problems over the following decades, his 2001 achievement remained unparalleled.
In 2022, Leng, then a second-year graduate student at UCLA, delved into Gowers’ theory. His goal was not to reprove the Seven Theorem but to tackle a technical