26/07/2023

Lecture on Ramsey Theorem draws praise

Entertaining, educational, and surprising. That's how the audience at the 34th Brazilian Mathematics Colloquium described the special lecture by Marcelo Campos (PhD from IMPA and postdoctoral researcher at the University of Oxford) and Simon Griffiths (PUC-Rio) on advances in the upper bound for Ramsey's Theorem.

The two researchers, along with Robert Morris (IMPA) and Julian Sahasrabudhe (Cambridge), stirred the world of mathematics earlier this year by sharing results on a new algorithm capable of improving the limit of the theorem. This advance, the most significant in the field since 1935, earned a special place at the 34th CBM.

The explanation of the results included a dialogue between Socrates, played by Simon, and a student tasked with answering questions, played by Marcelo. The dynamic elicited laughter from the students and researchers who packed the auditorium.

“Many students are coming to the colloquium for the first time, and we want people to get excited about combinatorics, for the community to grow, and it was for these people that we gave the lecture. So that everyone could understand it,” Campos stated.

Mariana Martins, a doctoral student in combinatorics at UFRJ, described the lecture as the highlight of the event.

"The presentation was excellent, the way they presented the results was extremely natural, and they are incredible. It's a great advancement for our field. It was exceptional and spectacular."

Matheus Manso, a doctoral student at IMPA, added that the lecture was very didactic. "Even though I know little about the subject, the explanation they provided was super clear; they were extremely didactic, especially for those unfamiliar with the topic. It was a great lecture."

Bruno Bandeira, a doctoral student in combinatorics at UFRJ, described the presentation as "fun, interesting, and also good entertainment."

Da esquerda para direita: Mariana, Bruno e Matheus

The last advance on the upper limit was in 1935.

The problem is interpreted using Graph Theory, where objects – called vertices – are interconnected by edges with blue or red colors. The sets are called "cliques," and the question is: what is the number of vertices needed to guarantee the existence of a clique of a certain size with edges of only one color?

In 1935, Erdős and Szekeres showed a pioneering result and arrived at the upper bound R(k)<4^k through an algorithm. Campos, Griffiths, Morris, and Sahasrabudhe found a new upper bound: (3.995)^k.

The result may contribute to research by experts in combinatorics and may even lead to developments in other areas.

Since 2018, the group has been meeting to explore the problem and, after many attempts, it was possible to define the best solution found. The "eureka" moment happened during the IMPA Summer Program, a unique opportunity for mathematicians to meet in person, since Julian Sahasrabudhe is an assistant professor at the University of Cambridge.

Previously, Sahasrabudhe was a distinguished postdoctoral fellow at IMPA (2017-2018), as was Simon Griffiths, also a distinguished postdoctoral fellow at the institute (2010-2013) and current adjunct professor at PUC-Rio. Marcelo Campos was supervised by Robert Morris during his doctorate, presented his thesis in March of this year, and went on to do postdoctoral work at Oxford.