Open. The site's label is OPEN (; page last edited 18 January 2026). Four pending partial claims are recorded. Two bound $f_k(N)$ without determining its order: Erdős's bound $f_k(N)\ll_k\log N/\log\log N$ of 1970 (claim page (Erdős, 1970)), and the bounds of Tang and Zhang of December 2025, $(\log N)^{c_k-o(1)}\le f_k(N)\ll(\log N)^{\mu_k^S-1+o(1)}$ with $\mu_k^S$ the sunflower-free capacity, together with their proof that $f_k(N)=(\log N)^{1-o(1)}$ exactly when the sunflower conjecture of Problem 857 fails at $k$ (claim page (Tang and Zhang, 2025)). Two later claims each assert $f_k(N)=(\log N)^{\gamma_k+o(1)}$ with an exponent defined by an extremal problem and not evaluated: a note of 15 April 2026 posted in the discussion thread, written with GPT-5.4 Pro, whose exponent is the infimum over $z>0$ of the growth rate of a weighted sunflower-free partition function minus $z$ (Chojecki's claim page (2026)); and a manuscript entered on the proof-claim tab on 18 July 2026 as a full claim, written with GPT 5.6 Sol Pro, whose exponent is the supremum of $(r/(en))M_k(n,r)^{1/r}$ over uniform families with no $k$ sets of equal pairwise union (the page of RayYoung, Zhu and Luo). These two are recorded as partial claims: the question asks for an estimate of $f_k(N)$, which for a function of polylogarithmic growth is its exponent, and each claim characterizes the exponent without evaluating it, its value left open on the claimants' own account (the note says that computing $\gamma_k$ remains open; the manuscript's authors tie it to the sunflower conjecture); what each covers is stated on its page. The site's curator restated the first of them in the thread without checking it; the second has no comment on the tab. None of the four claims has a journal record, and nothing is reviewed here. The standing in the frontmatter is open, derived from the pending partial claims, no full claim being recorded.