OPEN

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.

Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows
\[h(N) \ll N^{2/3}.\]
The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka [KeMe23] imply that
\[h(N) \gg \exp(c(\log N)^{1/12})\]
for some $c>0$.