Intersecting families with large shadow degree

A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial \mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of $\mathcal{F}$. The shadow degree (or minimum positive co-degree) of $\mathcal{F}$ is defined as the maximum integer $r$ such that every $E\in \partial \mathcal{F}$ is contained in at least $r$ members of $\mathcal{F}$. In 2021, Balogh, Lemons and Palmer determined the maximum size of an intersecting $k$-uniform family with shadow degree at least $r$ for $n\geq n_0(k,r)$, where $n_0(k,r)$ is doubly exponential in $k$ for $4\leq r\leq k$. In the present paper, we present a short proof of this result for $n\geq 2(r+1)^rk \frac{\binom{2k-1}{k}}{\binom{2r-1}{r}}$ and $4\leq r\leq k$.