Open (the site's label, which adds that the problem cannot be resolved by a finite computation). The sources below give separated lower and upper bounds at the exact host threshold $K_{2^n+1}$, but they do not determine the growth order; the search recorded under Current assessment found nothing further, and that negative result is not proof of openness. Two refereed bounds are accepted partial claims: Day and Johnson's lower bound, which answers Chung's question whether $f(n)\to\infty$, and Janzer and Yip's upper bound. Two partial claims are pending: Girão and Hunter's earlier upper bound, a preprint, and the value $f(3)=5$, recorded on its claim page (Cai, 2026).