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Mathematics of Computation

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Cutting planes for families implying Frankl's conjecture

Author: Jonad Pulaj
Journal: Math. Comp. 89 (2020), 829-857
MSC (2010): Primary 05D05, 90C10
Published electronically: July 25, 2019
MathSciNet review: 4044452
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Abstract: We find previously unknown families of sets which ensure Frankl's conjecture 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 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 Morris from 2006.

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Additional Information

Jonad Pulaj
Affiliation: Department of Mathematical Optimization, Zuse Institute Berlin (ZIB) Takustrasse 7, 14195 Berlin, Germany
Address at time of publication: Department of Mathematics and Computer Science, Davidson College, Davidson, North Carolina, 28035

Keywords: Frankl's conjecture, union-closed families, integer programming, cutting-plane method, extremal combinatorics
Received by editor(s): June 3, 2018
Received by editor(s) in revised form: February 20, 2019, and May 9, 2019
Published electronically: July 25, 2019
Additional Notes: The work for this article has been (partly) conducted within the Research Campus MODAL funded by the German Federal Ministry of Education and Research (BMBF grant number 05M14ZAM)
Article copyright: © Copyright 2019 American Mathematical Society