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arXiv · 2607.21589

Upper-shadow comparisons on the slice and the Frankl--Tokushige product conjectures

Fan Chang·Hong Liu·Miao Liu
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Abstract

Let $\mathcal{F}\subseteq\binom{[n]}k$, and let $\partial_{k\to\ell}\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing $\mu_k(\mathcal{F})=|\mathcal{F}|/\binom nk$, $p=k/n$, and $q=\ell/n$, we prove $$ \mu_\ell(\partial_{k\to\ell}\mathcal{F})\geq \begin{cases}\mu_k(\mathcal{F})^{\frac{\log q}{\log p}}, &0\leq \mu_k(\mathcal{F})\leq p^2,\\ q\left[1-\left(1-\frac{\mu_k(\mathcal{F})}{p}\right)^{ \frac{\log(1-q)}{\log(1-p)}} \ \ \right], &p^2\leq \mu_k(\mathcal{F})\leq p,\\ 1-(1-\mu_k(\mathcal{F}))^{\frac{\log(1-q)}{\log(1-p)}}, &p\leq \mu_k(\mathcal{F})\leq1. \end{cases} $$ This yields a dimension-free closed-form lower bound for the finite Kruskal--Katona profile and each branch is asymptotically sharp. As the main application, we settle both the uniform and biased product conjectures of Frankl and Tokushige. If $r\geq2$, $0\leq k_i\leq(r-1)n/r$, and $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ are $r$-cross-intersecting, then $$ \prod_{i=1}^r\mu_{k_i}(\mathcal{F}_i)\leq\prod_{i=1}^r\frac{k_i}{n}. $$ In biased product setting, we prove that for $r$-cross-intersecting families $\mathcal{A}_i\subseteq 2^{[n]}$ and $0\leq p_i\leq(r-1)/r$, $\prod_{i=1}^r\mu_{p_i}(\mathcal{A}_i)\leq\prod_{i=1}^rp_i.$ We further determine all equality cases in both settings.

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Fan Chang, Hong Liu, Miao Liu. 2026-07-23. Upper-shadow comparisons on the slice and the Frankl--Tokushige product conjectures. https://arxiv.org/abs/2607.21589

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