2023/06/01 by Van Magnan, Magnan, Van, Cory Palmer +3
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2306.00384
openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F⊆ \binom[n]r be an intersecting family of sets and let Δ(F) be the maximum degree in F, i.e., the maximum number of edges of F containing a fixed vertex. The diversity of F is defined as d(F) := |F| - Δ(F). Diversity can be viewed as a measure of distance from the `trivial' maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed, the diversity of this family is 0. Moreover, the diversity of the largest non-trivial intersecting family à la Hilton-Milner is 1. It is known that the maximum possible diversity of an intersecting family F⊆ \binom[n]r is \binomn-3r-2 as long as n is large enough. We introduce a generalization called the C-weighted diversity of F as dC(F) := |F| - C ⋅ Δ(F). We determine the maximum value of dC(F) for intersecting families F ⊆ \binom[n]r and characterize the maximal families for C∈ [0,(7)/(3)) as well as give general bounds for all C. Our results imply, for large n, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl's Delta-system method.