The site's wording quantifies over every set $X$, the empty set included; at $X=\varnothing$ its hypothesis reads $0<0$, which never holds, so no family meets it and the question as printed is answered yes for every choice of $c_t$, for want of an instance. The defect is Erdős's: [Er71] Problem 17 (p. 105) takes the count "for every $S_1\subset S$", and the site's curator, Thomas Bloom, quoted that sentence in the problem's thread on 4 December 2025 and called the site's statement an accurate rephrasing of it. Bloom reads the question with $X$ nonempty. The site's commentary (page last edited 25 January 2026) calls the statement false and names Wood's construction [Wo13b] and KoishiChan's as counterexamples, which refute it only once $X$ is nonempty; the Lean statement that Boris Alexeev posted in the thread on 22 January 2026, whose hypothesis is X.Nonempty, is the one Terence Tao recorded as the formalization of KoishiChan's solution and the one the label's (LEAN) mark refers to; and on 23 January 2026 Bloom wrote of marking the problem solved by KoishiChan "since this answers the question as I understand it". The corrected Statement inserts "nonempty" before "set $X$" and changes nothing else; the formal-conjectures statement requires $X$ nonempty as well. Under the printed wording the answer is yes, vacuously; under Bloom's reading it is no: Wood's theorem gives $c_t<2$ and KoishiChan's construction $c_t\le2$ for every $t$, and Lovász's theorem with [KN99] gives the exact value $1$ (Known Results). No result about the printed wording exists beyond the vacuity check recorded here. The site's label PROVED (LEAN) has the polarity of a positive answer; it contradicts the site's own commentary and the Lean proof of the negation, and the page follows the commentary (Status).