Ben Green presents super-polynomial lower bounds for off-diagonal van der Waerden numbers W(3,k): https://arxiv.org/abs/2102.01543, via https://gilkalai.wordpress.com/2021/02/08/to-cheer-you-up-in-difficult-times-20-ben-green-presents-super-polynomial-lower-bounds-for-off-diagonal-van-der-waerden-numbers-w3k/

W(3,k) is the smallest N such that a 2-coloring of [N] has a 3-term arithmetic progression of one color or a k-term progression of the other. It was previously known to be subexponential and thought to be only quadratic.