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 maximum 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≥ n₀(k,r), where n₀(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-1k2r-1r and 4≤ r≤ k.