相交并集族的卢贝尔界
The binomial norm of intersecting-union families
AI总结：
研究关于满足特定条件的族\(\mathcal{F}\subseteq 2^{[n]}\) 的卢贝尔界问题，核心方法是用\(p\)偏差测度框架取代循环置换论证并积分，主要贡献是证明弗兰克尔猜想并确定极值族，改进了圆法的估计。
AI中文摘要：
在2021年关于卡托纳圆法的综述中，弗兰克尔猜想，对于每个\(\mathcal{F}\subseteq 2^{[n]}\)的族，其中任意两个成员相交且没有两个成员覆盖\([n]\)，满足尖锐的卢贝尔型界\(\sum_{F\in \mathcal{F}}\binom{n}{|F|}^{-1}\le \frac{n + 1}{6}\)。这改进了圆法早期得到的估计\(\frac{n}{4}\)。本文证明了弗兰克尔猜想并确定了所有极值族。我们的证明用布尔格上的\(p\)偏差测度框架取代了循环置换论证，然后在整个概率范围内对所得估计进行积分。这种连续积分恢复了最优系数\(\frac{1}{6}\)，而圆法中固有的离散平均仅产生\(\frac{n}{4}\)。
英文摘要：
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 $ \lVert\mathcal F\rVert_n :=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1} \leq \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 $ μ_p(\mathcal F)+μ_{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.
