Open. The site's label is OPEN (page last edited 1 December 2025; so labeled on 2026-09-18 and 2026-10-06). The question asks for the order of $g_k(N)$, and no source determines it beyond the following: $g_3(N)=1$ for $N\ge3$ and $g_4(N)=3$ for $N\ge2$ (van Doorn 2026, Theorems 1 and 3, an arXiv preprint); $4\le g_5(N)<1.2\cdot10^8$ for $N\ge3$ (van Doorn 2026, Theorems 5 and 8), where the 1975 statement $g_5(N)\asymp\log N$ holds for the positive-integer variant $h_5$ only; $g_6(N)\asymp N^{1/2}$ (Choi, Erdős and Szemerédi 1975, Theorem 4, both bounds valid for the site's $g_6$); $g_k(N)\le2^{k-1}N^{1-2^{1-k}}$ for large $N$ (1975, Theorem 5) and $g_k(N)<4N^{1-2^{2-k}}$ for large $N$ (van Doorn 2026, Theorem 9, stated with a sketch); and $g_k(N)>N^{1-\epsilon}$ for all $k\ge k_0(\epsilon)$ and large $N$ (1975, Theorem 6). Open: the value of $g_5$ (bounded, between $4$ and $1.2\cdot10^8$), the constants for $k=6$, the order of $g_k$ for every $k\ge7$, and the exponent for large $k$. The results that settle instances of the question are recorded on the claim pages of Choi, Erdős and Szemerédi (accepted, partial: the order of $g_4$ and $g_6$ and the general bounds), van Doorn (claimed, partial: $g_3$ and $g_4$ exactly, $g_5$ bounded) and Erlbacher's release (claimed, partial: $g_5(N)\le3{,}519{,}219$, an AI-produced manuscript of 8 July 2026 with a Lean development, announced in the thread); none is a full claim, so the standing derived from them is open. No proof claim exists on the site. This is a bounded negative finding, not a certificate of openness.