Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq^{(k)}$ 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 …