Open. The site's label is OPEN (page last edited 4 April 2026). Six partial claims are recorded, none bearing on the displayed limit question. The refereed lower bounds the site credits are accepted partial claims: Berlekamp gives $f(k)\ge(\frac{\log2}{2}-o(1))k$ through his two-coloring bound $W(p+1)>p\,2^p$ for prime $p$, and Gerver gives $f(k)\ge(1-o(1))k\log k$. Among the numerical records, Wróblewski's set gives $f(3)\ge3.00849$, accepted on its journal record, and Walker's Kempner sets give $f(4)\ge4.43975$ and $f(10)\ge14.056$, an arXiv preprint that stays claimed. Corollary 11.2 of the OpenAI release manuscript Quasipolynomial bounds for arithmetic progressions (23 September 2026; claim page (OpenAI, 2026)) claims that $f(k)$ is finite for every $k\ge3$, with the bound $f(k)\le\sum_{m\ge0}2^{-m}r_k(2^m)$ and no numerical value. The finiteness itself follows, by Gerver's equivalence recorded in the site's commentary, from the reciprocal-sum theorem of Problem 3, which this corpus accepts on that problem's claim page (OpenAI, 2026); the explicit bound stays claimed. Neither gives an estimate of $f(k)$ or anything on the displayed limit question. Kiichi's explicit sets, posted on the problem's thread on 26 September 2026 with a manuscript of 3 October 2026, give $f(3)\ge3.0085385$ and $f(4)\ge4.4397534742$, beyond the records of Wróblewski and Walker; the claimants state each set as a Lean theorem of their own, which this corpus has not built, and claim nothing on the growth of $f(k)$. The release's companion manuscript Quantitative superexponential bounds for van der Waerden numbers (23 September 2026; intake card OpenAI 2026) proves $W(k)>k^{ck}$ with $c=10^{-5}$ for all large $k$, accepted with formalized evidence on Problem 138's claim page (OpenAI, 2026), hence $\log W(k)\ge ck\log k$; it makes no claim about $f(k)$ or the ratio $f(k)/\log W(k)$ and does not name this problem, so it is recorded here as an input to the limit question and gets no claim page. Gerver's lower bound and the trivial $f(k)/\log W(k)\ge1/2$ recorded by the site are unchanged by these results, so the problem stays open.