Open. The site's label is OPEN (page last edited 2 December 2025). Its commentary records the upper bounds $h(N)\ll N^{2/3}$, from a MathOverflow answer, and $h(N)\ll N^{\log3/\log22+o(1)}$ (an exponent of about $0.355$, which the site credits to Hunter's comment and the preprint of Shi and Dong below credits to the coloring of Deng, Tidor and Zhao, arXiv:2307.06914), and the lower bound $h(N)\gg\exp(c(\log N)^{1/9})$ for some $c>0$, which follows from Hunter's observation together with the bounds on sets without three-term progressions in [BlSi23] and [KeMe23]. The same observation applied to Raghavan's bound $r_3(N)\le N\exp(-c(\log N)^{1/6}/\log\log N)$ [Ra26] gives $h(N)\gg\exp(c(\log N)^{1/6-o(1)})$, a derivation posted as a comment on the problem's thread on 4 August 2026; as a thread post it has no claim page. Two partial claims on the site's proof-claims tab (as of 2026-10-06), neither adopted by the site, claim to lower the upper exponent: Itabe's bound $N^{1/3+o(1)}$ (credited to GPT-5.6, with a Lean development that is unbuilt and unaudited here) and Shi and Dong's bound $N^{1/4+o(1)}$ (arXiv:2607.20752, credited to GPT 5.6 Sol). Both are claimed and unreviewed; neither determines the order of $h(N)$, so the problem stays open.