Three mathematicians have set a new limit for when patterns show up in sets of integers, making it the first progress on one of the major unsolved problems in the field of combinatorics in decades.
In February this year, Ashwin Sah and Mehtaab Sawhney along with James Leng obtained a long-awaited improvement on an estimate of how larger sets of integers can get before they must contain sequences of evenly spaced numbers such as (9, 19, 29, 39, 49) or (30, 60, 90, 120), Quanta Magazine reported.
The proof contributes towards an extensive body of research on the mathematical impossibility of complete disorder. Further, it represents the first significant advancement on one of the largest unsolved issues in combinatorics in decades, it added.
Sah and Sawhney met as undergraduates at the Massachusetts Institute of Technology. Together, they have written 57 math proofs. Meanwhile, Leng is a graduate student at the University of California in Los Angeles.
Ben Green, a mathematician at the University of Oxford, has lauded their "phenomenally impressive" achievement. The three of them were pursuing graduation at the time their work was released.
Arithmetic progressions are the sequences of regularly spaced numbers. While they are simple patterns, they seemingly hide mathematical complexity and are difficult to avoid, sometimes impossible, too.
The report further highlights that mathematicians Paul Erdos and Pal Turan conjectured in 1936 that when a set consists of a nonzero fraction of whole numbers -- even when it is only 0.00000001% -- then it has to contain arbitrarily long arithmetic progressions. Only those sets can avoid arithmetic progressions that comprise a "negligible" portion of the whole numbers.
Another mathematician, named Endre Szemeredi, proved the conjecture in 1975, nearly 40 years later.
Szemeredi's work spawned multiple lines of research which are still being explored by mathematicians, who have built on his result in the context of finite sets of numbers.
If you try to avoid arithmetic progressions with four or more terms, this problem becomes tougher. "The thing I love about this problem is it just sounds so innocent, and it's not. It really bites," Sawhney said.
Sawhney has finished his PhD and is now a Clay Fellow at Columbia University, while Sah is still enrolled at MIT and is completing his graduation.
Their doctoral adviser at MIT, Yufei Zhao said Sawhney and Sah's "incredible strength is taking something that is extremely technically demanding and understanding it and improving upon it".
