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We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression.
On sets of integers which contain no three terms in arithmetical progression
Felix A Behrend · 1946
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Klaus F Roth · 1953
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Thomas F Bloom and Olof Sisask · 2019
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Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang · 2020
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Thomas F Bloom and Olof Sisask · 2020
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Mark Braverman, Subhash Khot, and Dor Minzer · 2022
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William T Gowers · 2023
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