2024/06/01 by Péter Frankl, Jian Wang, Frankl, Peter +1
Medicine · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.2406.00465
openalex publication_date 2024/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \frac\binom2k-1k\binom2r-1r and 4≤ r≤ k.