An improved lower bound for the union-closed sets conjecture by Will Sawin (2022 preprint, revised 2023)

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.