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PreprintCC BY 4.0arXiv

Beyond Liu's 0.382709 threshold for the union-closed sets conjecture

  • Simone Costa
  • Ankan Sadhu
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Abstract

The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let cUC denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold cL>0.382709087918735. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound cUC≥ cL. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain cUC≥ 0.3828852549667978. We also show that the inequality used in this proof cannot give a threshold larger than 0.3828852599667….

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Cite

BibTeXRIS
BibTeX
@misc{costa2026beyond,
  title = {{Beyond Liu's 0.382709 threshold for the union-closed sets conjecture}},
  author = {Simone Costa and Ankan Sadhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02295},
  eprint = {2610.02295},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02295},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Beyond Liu's 0.382709 threshold for the union-closed sets conjecture
AU  - Simone Costa
AU  - Ankan Sadhu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02295
UR  - https://arxiv.org/abs/2610.02295
AB  - The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let c_UC denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold c_L>0.382709087918735. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound c_UC≥ c_L. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain c_UC≥ 0.3828852549667978. We also show that the inequality used in this proof cannot give a threshold larger than 0.3828852599667….
ER  -