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Students Find New Evidence of the Impossibility of Complete Disorder
A new mathematic proof marks the first progress in decades on a problem about how order emerges.
With James Leng, a graduate student at UCLA, they obtained a long-sought improvement on an estimate of how big sets of integers can get before they must contain sequences of evenly spaced numbers, like {9, 19, 29, 39, 49} or {30, 60, 90, 120}. In 1936, the mathematicians Paul Erdős and Pál Turán conjectured that if a set consists of a nonzero fraction of the whole numbers—even if it’s just 0.00000001 percent—then it must contain arbitrarily long arithmetic progressions. For everything to work, they first had to strengthen an older, more technical result by Green, Terence Tao of UCLA and Tamar Ziegler of Hebrew University.
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