A local version of Katona's intersection theorem

Abstract

Katona's intersection theorem states that every intersecting family $\mathcal{F}\subseteq [n{]}^{(k)}$ satisfies $|\partial \mathcal{F}|\ge |\mathcal{F}|$, where $\partial \mathcal{F}=\{F\setminus x:x\in F\in \mathcal{F}\}$ is the shadow of $\mathcal{F}$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal{F}\subseteq [n{]}^{(k)}$, there is some $i\in [n]$ such that $|\partial \mathcal{F}(i)|\ge |\mathcal{F}(i)|$, where $\mathcal{F}(i)=\{F\setminus i:i\in F\in \mathcal{F}\}$ is the link of $\mathcal{F}$ at $i$. Here, we prove this conjecture in a very strong form for $n>\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$. In particular, our result implies that for any $j\in [k]$, there is a $j$-set $\{a_1,\dots ,a_j\}\in [n{]}^{(j)}$ such that $|\partial \mathcal{F}(a_1,\dots ,a_j)|\ge |\mathcal{F}(a_1,\dots ,a_j)|$. A similar statement is also obtained for cross-intersecting families.

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