Strong semimodular lattices and Frankl's conjecture

@article{Abe2000StrongSL,
  title={Strong semimodular lattices and Frankl's conjecture},
  author={Tetsuya Abe},
  journal={algebra universalis},
  year={2000},
  volume={44},
  pages={379-382}
}
  • Tetsuya Abe
  • Published 2000
  • Mathematics
  • algebra universalis
Abstract. In this paper, we show that Frankl's conjecture holds for strong semimodular lattices. The result is the first step to deal with the case of upper semimodular lattices. 
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References

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Frankl's Conjecture Is True for Lower Semimodular Lattices
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  • Mathematics, Computer Science
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Abstract. It is shown that every finite lower semimodular lattice L with |L|≥2 contains a join-irreducible element x such that at most |L|/2 elements y∈L satisfy y≥x.
Frankl's Conjecture is True for Modular Lattices
Abstract. It is shown that every finite modular lattice L with |L|≥2 contains a join-irreducible element x∈L such that at most |L|/2 elements y∈L satisfy y≥x.
Union-Closed Families
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TLDR
A number of equivalent conjectures are found and a general theorem stating exactly when a subfamily is enough to guarantee the existence of an element from the subfamily which is in half the sets of the whole family is proved. Expand