The binomial norm of intersecting-union families

Abstract

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 binomial norm bound $\Vert \mathcal{F}{\Vert }_n:={\sum }_{F\in \mathcal{F}}{\left(\genfrac{}{}{0ex}{}{n}{|F|}\right)}^{-1}\le \frac{n+1}{6}.$ 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 ${\mu }_p(\mathcal{F})+{\mu }_{1-p}(\mathcal{F}).$ Integrating this estimate over $p$ converts it directly into the desired binomial norm 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.

arXiv comment7 pages, the terminology and historical discussion have been revised following comments by Peter Frankl