Intersecting families with large shadow degree

A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$ for all $F,F^{\mathrm{\prime }}\in \mathcal{F}$. The shadow family $\mathrm{\partial }\mathcal{F}$ 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 $E\in \mathrm{\partial }\mathcal{F}$ 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 $n\ge n_0(k,r)$, where $n_0(k,r)$ is doubly exponential in $k$ for $4\le r\le k$. In the present paper, we present a short proof of this result for $n\ge 2(r+1{)}^{r}k\frac{\left(\genfrac{}{}{0px}{}{2k-1}{k}\right)}{\left(\genfrac{}{}{0px}{}{2r-1}{r}\right)}$ and $4\le r\le k$.