PROVED
This has been solved in the affirmative.
Can $\mathbb{N}$ be $2$-coloured such that if\[\{a,a+d,\ldots,a+(k-1)d\}\]is a $k$-term monochromatic arithmetic progression then $k\ll_\epsilon a^\epsilon$ for all $\epsilon>0$?
A question of Spencer, who proved that this is possible with $3$ colours, with $a^\epsilon$ replaced by a very slowly growing function $h(a)$ (the inverse of the van der Waerden function). Erdős reports that he can construct such a colouring with the bound $k\ll a^{1-c}$ for some absolute constant $c>0$. He knew no non-trivial lower bound.
Zach Hunter has proved the answer is yes (see the comments).
View the LaTeX source
Additional thanks to: Zach Hunter
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #984, https://www.erdosproblems.com/984, accessed 2025-11-15
