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Gil Kalai on recent developments in Roth's theorem: https://gilkalai.wordpress.com/2020/07/08/to-cheer-you-up-in-difficult-times-7-bloom-and-sisask-just-broke-the-logarithm-barrier-for-roths-theorem/

Salem and Spencer and later Behrend proved in the 1940s that subsets of [1,n] with no triple in arithmetic progression can have nearly linear size (see https://en.wikipedia.org/wiki/Salem%E2%80%93Spencer_set), and Klaus Roth proved in 1953 that they must be sublinear (https://en.wikipedia.org/wiki/Roth%27s_theorem_on_arithmetic_progressions). The upper bounds have slowly come down, to $n/{\log }^{1+c}n$ in this new result, but they're still far from Behrend's $n/e^{O(\sqrt{\log n})}$ lower bound.