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

Abstract

Let $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$, and let ${\partial }_{k\to \ell }\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing ${\mu }_k(\mathcal{F})=|\mathcal{F}|/(\genfrac{}{}{0ex}{}{n}{k})$, $p=k/n$, and $q=\ell /n$, we prove $${\mu }_{\ell }({\partial }_{k\to \ell }\mathcal{F})\ge \{\begin{array}{ll}{\mu }_k(\mathcal{F}{)}^{\frac{\log q}{\log p}},& 0\le {\mu }_k(\mathcal{F})\le p^2,\\ q\left[1-{\left(1-\frac{{\mu }_k(\mathcal{F})}{p}\right)}^{\frac{\log (1-q)}{\log (1-p)}}\right],& p^2\le {\mu }_k(\mathcal{F})\le p,\\ 1-(1-{\mu }_k(\mathcal{F}){)}^{\frac{\log (1-q)}{\log (1-p)}},& p\le {\mu }_k(\mathcal{F})\le 1.\end{array}$$ 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 $r\ge 2$, $0\le k_i\le (r-1)n/r$, and ${\mathcal{F}}_i\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k_i}\right)$ are $r$-cross-intersecting, then $$\prod _{i=1}^r{\mu }_{k_i}({\mathcal{F}}_i)\le \prod _{i=1}^r\frac{k_i}{n}.$$ In biased product setting, we prove that for $r$-cross-intersecting families ${\mathcal{A}}_i\subseteq 2^{[n]}$ and $0\le p_i\le (r-1)/r$, ${\prod }_{i=1}^r{\mu }_{p_i}({\mathcal{A}}_i)\le {\prod }_{i=1}^r{p}_i.$ We further determine all equality cases in both settings.

32 pages, a new version gives a clearer formulation of the upper-shadow comparisons on the slice and a complete characterization of all equality cases