Intersecting families with large shadow degree

A k-uniform family ℱ is called intersecting if F∩ F'≠∅ for all F,F'∈ℱ. The shadow family ∂ℱ is the family of (k-1)-element sets that are contained in some members of ℱ. The shadow degree (or minimum positive co-degree) of ℱ is defined as the minimum integer r such that every E∈∂ℱ is contained in at least r members of 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≥ n_0(k,r), where n_0(k,r) is doubly exponential in k for 4≤ r≤ k. In the present paper, we present a short proof of this result for n≥ 2(r+1)^rk 2k-1k/2r-1r and 4≤ r≤ k.