25 Park Place, Atlanta, GA

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Date and Time: 04/25/2024, 16:00--17:00

Location: 25 Park Place, Room 1441

Colloquium: Recent progress in Ramsey Theory

Speaker: Jacques Verstraete, University of California, San Diego

Title: Recent progress in Ramsey Theory

Abstract: The Ramsey number r(s,t) denotes the minimum N such that in any red-blue coloring of the edges of the complete graph K_N, there exists a red K_s or a blue K_t. While the study of these quantities goes back almost one hundred years, to early papers of Ramsey and Erdos and Szekeres, the long-standing conjecture of Erdos that r(s,t) has order of magnitude close to t^{s - 1} as t goes to infinity remains open in general.
It took roughly sixty years before the order of magnitude of r(3,t) was determined by Jeong Han Kim, who showed r(3,t) has order of magnitude t^2/(\log t). In this talk, we discuss a variety of new techniques which lead to the lower bound in the following statement: for some constants a,b > 0 and t \geq 2,
\[ a\frac{t^3}{(\log t)^4} \leq r(4,t) \leq b\frac{t^3}{(\log t)^2}.\]
This solves a conjecture of Erd\H{o}s. We also come close to determining related quantities known as Erdos-Rogers functions,
as well as determine the current best bounds for other graph Ramsey numbers.

Joint work in part with Sam Mattheus and Dhruv Mubayi.

Host: Yi Zhao (yzhao6@gsu.edu)


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