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