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Recent Advances in Ramsey Theory.

Mathematics Colloquium | Dhruv Mubayi | Jan 24th, 2023

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Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer s, there exists a positive integer n so that for every partition of the edges of the complete graph on n vertices into two classes, one of the classes must contain a complete subgraph on s vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest n that satisfies this property.

For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that “pseudo-randomness” and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

About Speaker : 

Dhruv Mubayi received his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 1998. After doing postdoctoral work at Georgia Tech and Microsoft Research, he joined the faculty at the University of Illinois at Chicago in 2002 where he is currently a Professor.  He  has served on the boards of several journals in discrete mathematics, including the Journal of Combinatorial Theory and the SIAM Journal on Discrete Mathematics.  His research has been funded continuously from the US National Science Foundation since 2004 and also currently by the Humboldt Foundation. He was awarded a Sloan Fellowship in 2005 and is a Fellow of the American Mathematical Society since 2019.

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