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By Will Sawin. Primary preprint arXiv:2211.11504, version3 (19 June 2023): arxiv.org/abs/2211.11504v3

Theorems1–2 and their proofs give the golden-ratio frequency bound through entropy growth for independent unions. The nontrivial-family application requires a nonempty union; singleton families containing a nonempty set satisfy the bound directly.

Let $\mathcal{F}$ be a finite union-closed family whose union is nonempty. Some ground element belongs to at least a proportionof its members. Requiring a nonempty union excludes both the empty set family and $\{\varnothing \}$, for which no element can be chosen.

Here is the entropy mechanism. Choose independent uniform random members $A,B$ of $\mathcal{F}$. Then $H(A)=\log |\mathcal{F}|$ and $H(A\cup B)\le \log |\mathcal{F}|$, because the union still belongs to $\mathcal{F}$. If every element had probability at most $u<c$ of belonging to $A$, Will Sawin's binary entropy estimate would givea contradiction when $|\mathcal{F}|>1$. A singleton set family with nonempty union satisfies the conclusion directly.^[ref]

Will Sawin works on problems connecting arithmetic and combinatorics. His paper on the union-closed sets conjecture applies entropy inequalities to a finite family closed under unions.^[ref]

For a nontrivial such family, the paper guarantees an element in at least a $(3-\sqrt{5})/2$ fraction of the sets. The argument studies the union of independent random members and bounds the resulting entropy growth. It is a quantitative advance toward the conjectured one-half fraction, with the nonempty-union condition needed to exclude the family containing only the empty set.^[ref]