Given a family $\mathcal{F}\subset 2^{[n]}$, 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.

中文翻译：

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多样性

给一个家庭 $\mathcal{F}\subset {\mathrm{2个}}^{[ñ]}$，其多样性是不包含具有最高程度的元素的集合的数量。事实证明，在k个均匀相交的家庭中，多样性的概念非常有用。在本文中，我们在其他极值集理论问题的背景下研究了多样性的（不同概念）。该论文的主要结果之一是，在多样性方面，交叉相交的族具有明显的稳定性，而在更普遍的意义上，Kruskal-Katona定理则具有明显的稳定性。我们还回答了黄先生关于不统一相交家庭的多样性的问题。