The Lubell bound for intersecting-union families

In a 2021 survey on Katona’s circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp Lubell-type bound ${\sum }_{F\in \mathcal{F}}{\left(\genfrac{}{}{0px}{}{n}{\mathrm{\mid }F\mathrm{\mid }}\right)}^{-1}\le \frac{n+1}{6}\mathrm{.}$ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl’s conjecture and determine all extremal families. Our proof develops a continuous $p$-biased measure approach in place of the circle method. The intersection and union conditions lead to a sharp estimate for ${\mu }_p(\mathcal{F})+{\mu }_{1-p}(\mathcal{F})\mathrm{.}$ Integrating this estimate over $p$ converts it directly into the desired Lubell bound and recovers the optimal coefficient $\frac{1}{6}$. This continuous averaging is the key new ingredient of the proof and also yields the characterization of all extremal families.