![]() “But it wasn’t something that was at the forefront of my research. ![]() “Many people have thought about r(4, t)-it’s been an open problem for over 90 years,” Verstraete said in a press statement. Scientists Solve 50-Year-Old Möbius Mystery.But the technique hit a brick wall when applied to the notoriously tricky r(4, t) problem, so Verstraete ventured into other areas of mathematics-specifically, finite geometry-to figure out the answer. Using a concept known as pseudorandom graphs, Verstaraete discovered a solution for the r(3, t) in 2019. These problems are usually solved using random graphs, but this Ramsey problem required more out-of-the-box thinking. Verstraete first saw the problem stated in a book about Erdös written by two UC San Diego researchers. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?” “This is what Sam and I have achieved in our recent work. “Because these numbers are so notoriously difficult to find, mathematicians look for estimations,” Verstraete said in the press statement. However, Verstraete and UC San Diego researcher Sam Mattheus have found an estimated solution for r(4, t), where t means that “points without lines” is variable. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”Įxpanding this idea, mathematicians Paul Erdös and George Szekeres discovered that r(4,4) equals 18 in 1935, and the solution to r(5,5) is still unknown. “It doesn’t matter what the situation is or which six people you pick-you will find three people who all know each other or three people who all don't know each other. “It’s a fact of nature, an absolute truth,” Jacques Verstraete, whose groundbreaking finding is currently under review with the journal Annals of Mathematics, says in a press release. (While I’m sure you have five close friends you can invite to a party, we’re talking about points and lines on a graph here-not actually people.) This is expressed as r(3,3), with the answer being six. While this is enough to make a non-mathematician dizzy, the concept is usually described as “the theorem on friends and strangers,” and is often allegorically explained as the idea that a party of six people will inevitably produce three people who know each other and three people who don’t. AI Is Helping Build the 'Periodic Table of Shapes".This is expressed as r(s, t), where s means “points with lines” and t means “points without lines.” Chiefly, the theorem states that a set of points will have no lines between them, or a set of points with all possible lines between them (aka cliques). Ramsey’s original theory says that if a graph is large enough-in mathematics, a graph is a collection of points and lines between those points-you can find order within all that chaos. Named after mathematician Frank Ramsey, who first proved his eponymous theorem in the 1920s, Ramsey problems seek to find order in disorder. ![]() However, after 90 years left unsolved, mathematicians from the University of California San Diego have pushed the upper limit and found a solution for the problem 4(r, t). Like many great mathematical conundrums, Ramsey problems are easy to understand but devilishly difficult to solve.
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