Morris's asymptotic conjecture: $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$

It is proved that $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ for every fixed $k\ge2$, resolving Morris's Conjecture 2. The proof combines robust weighted certificates, pair lifting and extremal forcing with a self-contained lower-bound argument. For four-sets, $n^2/4-O(n)\le\mathrm{FC}(4,n)\le(18+o(1))n^2$. More generally, for fixed $k\ge4$, $\mathrm{FC}(k,n)\ge\frac{n^{k-2}}{2(k-2)!}-O_k(n^{k-3})$. The sharp FC and robustness thresholds for sunflowers with a two-point core and disjoint pair petals are nine and ten petals. Exact finite certificates establish $\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture.

It is proved that $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ for every fixed $k\ge2$, resolving Morris's Conjecture 2. The proof combines robust weighted certificates, pair lifting and extremal forcing with a self-contained lower-bound argument. For four-sets, $n^2/4-O(n)\le\mathrm{FC}(4,n)\le(18+o(1))n^2$. More generally, for fixed $k\ge4$, $\mathrm{FC}(k,n)\ge\frac{n^{k-2}}{2(k-2)!}-O_k(n^{k-3})$. The sharp FC and robustness thresholds for sunflowers with a two-point core and disjoint pair petals are nine and ten petals. Exact finite certificates establish $\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture.