Undergraduate Student Makes Breakthrough in Graph Theory

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Source: Quanta Magazine

Sophie Zhu

In May, the then-20-year-old undergraduate student Ashwin Sah at Massachusetts Institute of Technology made a breakthrough in the field of graph theory. 

Having grown up in Portland, Oregon, Ashwin Sah displayed an astonishing talent in competitive and research mathematics. At the mere age of 16, he won the United States a gold medal at the International Mathematical Olympiad. He enrolled in MIT as an undergraduate, then graduated only 2.5 years later. However, even as an undergraduate, he immediately started off his freshman year of college by attending graduate-level seminars and courses. Specifically, he took two classes with the MIT professor Yufei Zhao, who, having taken note of the student’s talent, became a mentor to Sah’s research. Sah worked closely with a friend named Mehtaab Sawhney; the two strenuously performed research in combinatorial and probabilistic topics in mathematics, which was a suitable field for them as relatively little background knowledge was needed. Rather, an innovative and creative mind was, fitting their competition math experiences. Zhao notes that the two were always eager and motivated to embark on their questions, and he had led them through the paper-writing and research question process. 

Sah’s work focused on Ramsey numbers, a central tool in combinatorics that quantifies how large a graph must be to contain a certain kind of structure. (A graph is a set of vertices connected by edges.) Usually these “structures” can pertain to whether or not one can color the edges of the graph under specific conditions. For instance, given 6 vertices with 15 edges (every vertex connected to another), no matter the coloring of the edges (given red or blue), one will always find 3 vertices connected to each other with edges of the same color (called a clique). However, given 5 vertices and 10 edges, this does not hold true. 

For small numbers, this procedure may seem uninteresting and a sheer matter of brute force. However, as the size of the clique one seeks increases, this process becomes incredibly difficult. So, graph theorists seek to find upper and lower bounds for Ramsey numbers, which helps narrow the search for them. In the 1930s, mathematicians Paul Erdos and George Szekeres set off the study for these bounds, but little progress was made by graph theorists after that. Ashwin Sah improved the upper bound for 2-color Ramsey numbers by tweaking Erdos’ and Szekeres’ methods. 

Sah and Sawhney, now first-year graduate students at MIT, jointly won the 2021 Morgan Prize, one of the most prestigious nation-wide prizes to recognize the best undergraduate research. The two form an inseparable pair of mathematicians, constantly contacting each other to further their work.

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