A generalization of diversity for intersecting families

Let F⊆ [Formula presented] 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 Erdős–Ko–Rado Theorem. Indeed, the diversity of this family is 0. Moreover, the diversity of the largest non-trivial intersecting family, due to Hilton–Milner, is 1. It is known that the maximum possible diversity of an intersecting family F⊆[Formula presented] is [Formula presented] as long as n is large enough. We introduce a generalization called the C-weighted diversity of F as d_C(F)≔|F|−C⋅Δ(F). We determine the maximum value of d_C(F) for intersecting families F⊆[Formula presented] 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.