Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture

We verify an explicit inequality conjectured in [Gilmer, 2022, arXiv:2211.09055], thus proving that for any nonempty union-closed family F ⊆ 2[n], some i∈ [n] is contained in at least a 3-√52 ≈ 0.38 fraction of the sets in F \. One case, an explicit one-variable inequality, is checked by computer calculation.