We finally know how big a set of numbers can get before it has to contain a pattern known as a “polynomial progression.” From a report: A new proof by Sarah Peluse of the University of Oxford establishes that one particularly important type of numerical sequence is, ultimately, unavoidable: It’s guaranteed to show up in every single sufficiently large collection of numbers, regardless of how the numbers are chosen. “There’s a sort of indestructibility to these patterns,” said Terence Tao of the University of California, Los Angeles. Peluse’s proof concerns sequences of numbers called “polynomial progressions.” They are easy to generate — you could create one yourself in short order — and they touch on the interplay between addition and multiplication among the numbers. For several decades, mathematicians have known that when a collection, or set, of numbers is small (meaning it contains relatively few numbers), the set might not contain any polynomial progressions. They also knew that as a set grows it eventually crosses a threshold, after which it has so many numbers that one of these patterns has to be there, somewhere. It’s like a bowl of alphabet soup — the more letters you have, the more likely it is that the bowl will contain words. But prior to Peluse’s work, mathematicians didn’t know what that critical threshold was. Her proof provides an answer — a precise formula for determining how big a set needs to be in order to guarantee that it contains certain polynomial progressions. Previously, mathematicians had only a vague understanding that polynomial progressions are embedded among the whole numbers (1, 2, 3 and so on). Now they know exactly how to find them.
of this story at Slashdot.
Source:: Slashdot