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-1{k}2r-1{r} and 4 r k.