w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
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The Jamneshan-Tao conjecture holds for finite abelian groups of rank at most R via an inverse theorem linking non-trivial Gowers norms to bounded-complexity nilsequences.
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Three-color van der Waerden numbers grow super-exponentially
w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
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The Jamneshan-Tao conjecture for finite abelian groups of bounded rank
The Jamneshan-Tao conjecture holds for finite abelian groups of rank at most R via an inverse theorem linking non-trivial Gowers norms to bounded-complexity nilsequences.