After 88 years, mathematicians advance in ‘party theorem’ – 06/05/2023 – Science

An event planner has a mission to throw a party where 30% of the guests know each other or 30% don’t know each other. How many people does he have to send the invitation to?

The example is a way of presenting the theorem proposed by the British Frank Plumpton Ramsey (1903-1930), according to which it is always possible to find a certain type of pattern in a large enough structure.

Or, in the party example, that there are a finite number of people (R) to be invited to k guests know each other or don’t know each other.

The proposition has challenged mathematicians around the world for decades and, recently, earned researchers working in Brazil a mention in the journal Nature and praise from William Timothy Gowers, winner of the Fields Medal, known as the “Nobel of mathematics”, in 1998.

“Basically every combinatorics researcher has tried very hard to answer this question, including myself,” wrote Gowers on his Twitter profile. “Congratulations to Marcelo, Simon, Rob and Julian”.

The Brazilian Marcelo Soares Campos, the Englishmen Simon Griffiths and Robert David Morris and the Canadian Julian Sahasrabudhe advanced in what mathematicians call the upper limit, something that had not happened since 1935, when the Hungarians Paul Erdős and George Szekeres established conditions for the Theorem of Ramsey.

Erdős and Szekeres showed that, in order to have k people in the party who know each other or not, it is necessary to have at most 4^{kguests. Now, scientists have concluded that the maximum is 3,995k.}

The 0.005 change may seem insignificant, but its consequences could stimulate further research with the theorem and impact computer science, an area closely linked to combinatorics.

“More people are going to work on this problem and we will get a better understanding of this type of structure”, says Campos, who concluded his doctorate at IMPA (Institute of Pure and Applied Mathematics) in Rio de Janeiro in March, and in June embarks on a postgraduate -PhD at Oxford University (UK).

“A theorem is a mathematical truth that we deduce from certain very basic positions, with which anyone would agree”, explains Campos.

The challenge is that the results do not always match this truth. Or else it seems that they match, but there by the trillions, failures begin to emerge. If there were the theorem “every odd number is a prime number”, for example, the first answers would confirm the idea because, in fact, 1, 3, 5 and 7 are prime. But 9, which can be divided not only by 1 and itself but also by 3, would refute the proposal.

The reasoning developed in this process of trying to prove a theorem can be used in other areas, such as physics, for problem solving.

“To prove that the proposal is true, we need to show that it is valid for any configuration, and in our group we imagine that we are dealing with an adversary. He chooses the most difficult configurations and we have to verify”, says Campos.

The adversary has occupied the imagination of researchers since 2018, when Sahasrabudhe, currently a professor at the University of Cambridge, was doing postdoctoral work at IMPA and began working with Morris, a professor at the institute, and Griffiths, a professor at PUC-Rio.

The trio formulated numerous proposals. Sometimes, it seemed that everything was going well, but the invisible opponent came and triumphed again. In 2021, Campos joined the group and the battle continued.

The scientists held meetings to discuss the problem and, at the beginning of each year, they met in person at the Impa summer course to spend the weeks in front of a blackboard, using chalk as a weapon.

“Only now, in 2023, have we found a strategy that really worked”, says Campos.

At first, the group did not believe in the result. After so many defeats, they were sure that their rival would find a breach in their line of reasoning. “It’s a feeling of apprehension, that things could fall apart, that they could find a mistake at any moment”, describes the Brazilian.

Confidence came test after test. After days of checking, they began to believe that they had, in fact, achieved a breakthrough in Ramsey’s Theorem.

The process will be submitted to one of the largest scientific journals in the area of ​​mathematics and has been shared in congresses. In one, Gowers was in the audience and later confessed to being wary of feelings like “if I had tried a little harder I could have solved this problem myself”. But he felt the opposite.

“The evidence was very different from the argument I was trying to make, and I don’t think I would have found it. So, even if a little dream died, I am very happy to see this important discovery,” he wrote.