Formalizing Frankl’s Conjecture: FC-Families

  • Filip Marić
  • Miodrag Živković
  • Bojan Vučković
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7362)

Abstract

The Frankl’s conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every union-closed family containing them satisfies the Frankl’s condition (e.g., in every union-closed family that contains a one-element set a, the element a is contained in at least half of the sets, so families of the form a are the simplest FC-families). FC-families play an important role in attacking the Frankl’s conjecture, since they enable significant search space pruning. We present a formalization of the computer assisted approach for proving that a family is an FC-family. Proof-by-computation paradigm is used and the proof assistant Isabelle/HOL is used both to check mathematical content, and to perform (verified) combinatorial searches on which the proofs rely. FC-families known in the literature are confirmed, and a new FC-family is discovered.

Keywords

Weight Function Java Program Proof Assistant Semimodular Lattice Combinatorial Search 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Filip Marić
    • 1
  • Miodrag Živković
    • 1
  • Bojan Vučković
    • 1
  1. 1.Faculty of MathematicsUniversity of BelgradeSerbia

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