Frankl-complete sunflowers and extremal families of four-sets

A finite configuration is Frankl-complete if every union-closed extension has an element of the original support in at least half of its members. We prove that a sunflower consisting of a two-point core and disjoint two-point petals is Frankl-complete if and only if it has at least nine petals. The positive direction follows from a charging inequality valid for every admissible family, with weight three on the core and weight one on the petals. Combining this criterion with an elementary extremal estimate gives FC(4,n) = Θ(n²), resolving the four-uniform case of Morris’s asymptotic conjecture. Independently, we establish FC(4,9) = 16 by an explicit fifteen-block negative certificate and a computer-assisted exhaustive upper bound. A smaller positive certificate also refutes a lexicographic extremality conjecture of Pulaj and Wood. We give further finite bounds and explain why higher-uniformity sunflowers cannot provide the same local forcing mechanism.

Version V1. Code and exact certificates are available in the companion repository: https://github.com/michaeliu4/frankl-complete-four-sets/releases/tag/V1