Diversity

11/02/2018

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by   Peter Frankl, et al.

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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.