On the $C$-diversity of intersecting hypergraphs

TL;DR

This paper investigates the $C$-diversity of intersecting hypergraphs, establishing bounds and stability results for families with certain intersection properties, and characterizes extremal configurations for specific parameters.

Contribution

It introduces bounds on $C$-diversity for intersecting hypergraphs and characterizes extremal families, extending classical results with new stability insights.

Findings

For $1< C<rac{3}{2}$, the maximum $C$-diversity is achieved by a specific family $_{123}$.

A strong stability result is proven for the case $C=1$ (ordinary diversity).

The paper provides conditions on $n$ relative to $k$ for the bounds to hold.

Abstract

Let $\mathcal{F}\subset \binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. Let $\Delta(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\geq 1$ the $C$-diversity, $\gamma_C(\mathcal{F})$ is defined as $|\mathcal{F}|-C\Delta(\mathcal{F})$. Define $\mathcal{F}_{123} =\left\{F\in \binom{X}{k}\colon |F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\binom{n-3}{k-2}$. The main result shows that for $1< C<\frac{3}{2}$ and $n\geq \frac{42}{3-2C}k$, $\gamma_C(\mathcal{F})\leq \gamma_C(\mathcal{F}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to $\mathcal{F}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven.