Characterizing 3-Sets in Union-Closed Families

Let [n]:={1,2,…,n} and let a k-set denote set of cardinality k. A family sets is union-closed (UC) if the union any two in also family. Frankl’s conjecture states that for nonempty UC F⊆2[n] such F≠{∅}, there exists an element i∈[n] contained at least half F, where 2[n] denotes power on [n]. The 3-sets Morris smallest number distinct (whose n-set) ensure satisfied (in contains them ⌊n/2⌋+1 all n≥4. For A⊆2[n], Poonen’s theorem characterizes existence weights [n] which families contain satisfy conjecture, however determination specific nontrivial even small n. We classify n≤9 using polyhedral interpretation exact rational integer programming. This yields proof conjecture.