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arXiv · 2609.08291

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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Yunjiang Jiang. 2026-09-08. Entropy bounds and global couplings for union-closed families. https://arxiv.org/abs/2609.08291

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