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PreprintCC BY 4.0arXiv

Entropy bounds and global couplings for union-closed families

  • Yunjiang Jiang
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Abstract

We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026entropy,
  title = {{Entropy bounds and global couplings for union-closed families}},
  author = {Yunjiang Jiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08291},
  eprint = {2609.08291},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08291},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Entropy bounds and global couplings for union-closed families
AU  - Yunjiang Jiang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08291
UR  - https://arxiv.org/abs/2609.08291
AB  - We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.
ER  -