Diversity

Abstract

Given a family F⊂2[n], its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of k-uniform intersecting families. In this paper, we study (different notions of) diversity in the context of other extremal set theory problems. One of the main results of the paper is a sharp stability result for cross-intersecting families in terms of diversity and, slightly more generally, sharp stability for the Kruskal–Katona theorem. We also answer a question of Huang on the diversity of non-uniform intersecting families.