A Local Version of Katona’s Intersecting Shadow Theorem

Abstract

Katona’s intersection theorem states that every intersecting family $\mathcal {F}\subseteq [n]^{(k)}$ satisfies $\vert \partial \mathcal {F}\vert \geqslant \vert \mathcal {F}\vert$, 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 $\vert \partial \mathcal {F}(i)\vert \geqslant \vert \mathcal {F}(i)\vert$, 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 $n> \left( {\begin{array}{c}k+1\\ 2\end{array}}\right)$. 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 $\vert \partial \mathcal {F}(a_1,\dots ,a_j)\vert \geqslant \vert \mathcal {F}(a_1,\dots ,a_j)\vert$. A similar statement is also obtained for cross-intersecting families.