Mathematicians Crack a Century-Old Problem That's Perfect for Your Next Party
hubie writes:
Mathematicians have found a new way to impose order on chaos in the form of an answer to a challenge which has puzzled them for nearly a century:
In mathematics, Ramsey theory deals with the 'order in disorder'. No matter how complex a large system is, order will emerge as a smaller subsystem with unique structure.
Humans are pattern-seeking creatures living in a world of random chaos. We look for order in everything, from our lives, the world around us, to the Universe, and you could say Ramsey theory explains our ability to find it.
Ramsey numbers can be thought of as representing the boundaries of disorder. And it's notoriously hard to figure them out.
Since mathematician Frank Ramsey proved Ramsey's Theorem in the late 1920s, there's been perplexion on the specific problem that Sam Mattheus and Jacques Verstraete of the University of California, San Diego, finally cracked.
"Many people have thought about r(4,t) - it's been an open problem for over 90 years," Verstraete says.
"It really did take us years to solve. And there were many times where we were stuck and wondered if we'd be able to solve it at all."
[...] A common analogy for Ramsey theory requires us to consider how many people to invite to a party so that at least three people will either already be acquainted with each other or at least three people will be total strangers to each other.
Here, the Ramsey number, r, is the minimum number of people needed at the party so that either s people know each other or t people don't know each other. This can be written as r(s,t), and we know the answer to r(3,3) = 6.
"It's a fact of nature, an absolute truth. 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," Verstraete says.
"You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other."
[...] After almost a year and several math obstacles, they found r(4,t) is close to a cubic function of t. For a party with four people who all know each other or t people who don't, you need t3 people.
[...] "One should never give up, no matter how long it takes," Verstraete says. "If you find that the problem is hard and you're stuck, that means it's a good problem."
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