Open. The site's label is OPEN (page last edited 8 April 2026; site export of 2026-10-06); its commentary records the trivial $G_k(N)\le R_k(N)$, that the inequality can be strict ($G_3(5)=3$ against $R_3(5)=4$), and the theorem of Komlós, Sulyok and Szemerédi [KSS75] that $R_k(N)\ll_kG_k(N)$. No claim page is recorded. Theorem 1.1 of the OpenAI release manuscript Quasipolynomial bounds for arithmetic progressions (23 September 2026) claims $R_k(N)\le C_kN\exp(-c_k(\log N)^{\varepsilon_k})$ for every fixed $k\ge3$; it bounds $G_k(N)$ from above only through the trivial inequality $G_k(N)\le R_k(N)$ and settles none of the problem's three questions, the size of $G_k(N)$, its comparison with $R_k(N)$ and the limit of $R_3(N)/G_3(N)$, so it has no claim page, and the problem is open with no claim.