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  <doc>
    <id>5680</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2015-12-15</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">New Conjectures For Union-Closed Families</title>
    <abstract language="eng">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&#13;
wide-reaching implications if proven true. Finally, we partially prove the new conjectures and discuss possible approaches to solve them completely.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-56804</identifier>
    <author>Jonad Pulaj</author>
    <submitter>Jonad Pulaj</submitter>
    <author>Annie Raymond</author>
    <author>Dirk Theis</author>
    <series>
      <title>ZIB-Report</title>
      <number>15-57</number>
    </series>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="institutes" number="tele">Mathematics of Telecommunication</collection>
    <collection role="persons" number="pulaj">Pulaj, Jonad</collection>
    <collection role="projects" number="no-project">no-project</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5680/ZIB.pdf</file>
  </doc>
</export-example>
