A generalization of diversity for intersecting families

Let $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ be an intersecting family of sets and let $\Delta \left(F\right)$ 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 $d\left(F\right)≔\left|F\right|-\Delta \left(F\right)$. 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 $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ is $\left(\frac{n-3}{r-2}\right)$ as long as $n$ is large enough.

We introduce a generalization called the $C$-weighted diversity of $F$ as $d_C\left(F\right)≔\left|F\right|-C\cdot \Delta \left(F\right)$. We determine the maximum value of $d_C\left(F\right)$ for intersecting families $F\subseteq \left(\frac{\left[n\right]}{r}\right)$ and characterize the maximal families for $C\in \left(0,\frac{7}{3}\right)$ 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.