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 \(\mathcal{F}\). Balogh, Lemons and Palmer [1] 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 \(n\geq 2\frac{(r+1)^r}{\binom{2r-1}{r}}k\binom{2k}{k-1}\) and \(4\leq r\leq k\).