On the C-diversity of intersecting hypergraphs

2023/08/27 by Péter Frankl, Jian Wang, Frankl, Peter +1 · 1 citation
Social Sciences · #Combinatorics (math.CO) #FOS: Mathematics #Japanese History and Culture

paper · pdf · doi:10.48550/arxiv.2308.14028

openalex publication_date 2023/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F⊂ \binomXk be a family consisting of k-subsets of the n-set X. Suppose that F is intersecting, i.e., F∩ F'≠ ∅ for all F,F'∈ F. Let Δ(F) be the maximum degree of F. For a constant C≥ 1 the C-diversity, γ_C(F) is defined as |F|-CΔ(F). Define F₁₂₃ =\F∈ \binomXk\colon |F∩ \1,2,3\|=2\. It has C-diversity (3-2C)\binomn-3k-2. The main result shows that for 1< C<(3)/(2) and n≥ (42)/(3-2C)k, γ_C(F)≤ γ_C(F₁₂₃) with equality if and only if F is isomorphic to F₁₂₃. For the case of ordinary diversity (C=1) a strong stability is proven.

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• On the largest degrees in intersecting hypergraphs

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