New Conjectures For Union-Closed Families

Please always quote using this URN: urn:nbn:de:0297-zib-56804
  • The Frankl conjecture, also known as the union-closed sets conjecture, states that there exists an element in at least half of the sets of any (non-empty) union-closed family. From an optimization point of view, one could instead prove that 2a is an upper bound to the number of sets in a union-closed family with n elements where each element is in at most a sets, where a and n are non-negative integers. Formulating these problems as integer programs we observe that computed optimal values do not vary with n. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we partially prove the new conjectures and discuss possible approaches to solve them completely.

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  • Submitted for publication to refereed international journal

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Author:Jonad Pulaj, Annie Raymond, Dirk Theis
Document Type:ZIB-Report
Date of first Publication:2015/12/15
Series (Serial Number):ZIB-Report (15-57)