Theoretical analysis resolves Morris's asymptotic conjecture across all uniformities, establishing exact scaling bounds for Frankl-complete hypergraph configurations.
It is proved that \(FC(k,n)=Θ_k(nᵏ⁻²)\) for every fixed \(k≥2\), resolving Morris’s asymptotic conjecture. The new upper bound combines robust pair lifting of weighted certificates with extremal forcing, matching Morris’s lower bound. Sharp four-uniform sunflower thresholds are obtained, while exact finite certificates establish \(FC(4,9)=16\) and refute a lexicographic extremality conjecture of Pulaj and Wood. The general asymptotic proof is independent of the finite classification. V2 extends the four-uniform result of V1 to all fixed uniformities, retaining its sharp and finite results. Code and exact certificates are available in the companion repository: https://github.com/michaeliu4/frankl-complete-four-sets/releases/tag/V1
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Liu Mingchang (2026) studied this question.
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