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Cutting planes for families implying Frankl’s conjecture. (English) Zbl 1429.05199
Summary: We find previously unknown families of sets which ensure Frankl’s conjecture [P. Frankl, Extremal set systems, in: Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland); Cambridge, MA: MIT Press. 1293–1329 (1995; Zbl 0844.05094)] holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty finite union-closed (UC) family there exists an element of the ground set in at least half the sets of the considered UC family. Poonen’s theorem [B. Poonen, J. Comb. Theory, Ser. A 59, No. 2, 253–268 (1992; Zbl 0758.05096)] characterizes the existence of weights which determine whether a given UC family implies the conjecture for all UC families which contain it. We design a cutting-plane method that computes the explicit weights which satisfy the existence conditions of Poonen’s theorem. This method enables us to answer several open questions regarding structural properties of UC families, including the construction of a counterexample to a conjecture of R. Morris [Eur. J. Comb. 27, No. 2, 269–282 (2006; Zbl 1082.05092)].
05D05 Extremal set theory
90C10 Integer programming
Full Text: DOI arXiv
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