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A local version of Katona's intersection theorem

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