Last March 16, three seminars discussing the same mathematical theorem were delivered almost simultaneously at three different research centers. Having obtained his PhD less than a week prior at the Institute for Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Marcelo Campos of Brazil addressed an audience of about 20 people at the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP). Simon Griffiths, a British professor at the Pontifical Catholic University of Rio de Janeiro (PUC-RJ), gave a presentation at IMPA, while Julian Sahasrabudhe from Canada discussed the topic at the Faculty of Mathematics, University of Cambridge, in the UK.

Later that same day, following the conclusion of the seminars, the trio of mathematicians, along with their British colleague Rob Morris, a researcher at IMPA, jointly uploaded a four-authored paper to arXiv, a repository for preprints, or papers not yet peer-reviewed. A portion of the research community in the field of combinatorics — a branch of mathematics that investigates the extremal, probabilistic, and structural properties of finite objects — had already seen the paper. “Basically all combinatorialists have tried hard to answer this question — including me — and I think it’s fair to say that it is one of the top two or three open problems in extremal combinatorics, or perhaps even the actual top one,” tweeted British mathematician Timothy Gowers — a recipient of the Fields Medal, one of the most prestigious awards in mathematics — a day after he attended the seminar in Cambridge.

As its title suggested, the paper represented an “exponential improvement” on the Ramsey theorem, symbolized by the notation *R(k)*. The theorem itself carries a deceptively simple statement, yet since 1935, there has been little to no substantial progress towards its resolution. “We haven’t actually solved the theorem,” Campos explains. “What we’ve achieved is an algorithm that effectively reduces the upper bound on the solution.”

To a layperson this might appear modest, but what the researchers accomplished received global acclamation as a significant feat, with their paper receiving extensive coverage in science magazines and in the broader media. “I was floored” on learning about the paper, remarked mathematician Yuval Wigderson from Tel Aviv University, in an interview with *Quanta Magazine*. “I was literally shaking for half an hour to an hour.” Guilherme Mota, a researcher from IME-USP who organized the seminar given by Campos at the São Paulo university, was also surprised by the contributions of his colleagues. “Every expert in combinatorics has tackled this theorem at some point,” says Mota, who also works in the field and has collaborated with two of the four authors of the new paper. “After such a long time without major breakthroughs, it was unexpected.”

Formulated in 1930 by British mathematician Frank Plumpton Ramsey (1903–1930), the theorem that bears his name is as easy to understand as it is difficult to solve as the integer value in its formula increases. A popular adaptation of its formal statement poses the question in everyday terms: *what is the minimum number of guests a party must have to ensure there are at least a given number of mutual acquaintances or at least the same number of mutual strangers?* It is not necessary that both conditions are satisfied. This mysterious number of guests is represented generically by the letter *k *in the Ramsey theorem, denoted as *R(k)*. In this example, it is assumed that the relationship of acquaintance or nonacquaintance is reciprocal: if one person is another person’s acquaintance, then the latter is also the former’s acquaintance.