On the maximum diversity of hypergraphs with fixed matching number

Abstract

Let $\left[n\right]=\{1,2,\dots ,n\}$ and let $F$ be a family of $k$-subsets of $\left[n\right]$. The matching number of $F$ is defined as the maximum number of pairwise disjoint members in $F$. The covering number of $F$ is defined as the minimum size of $T\subset \left[n\right]$ such that each member of $F$ intersects $T$. Define the $t$-diversity of $F$ as the minimum size of $F^{\prime }$ such that $F\setminus F^{\prime }$ has covering number $t$. Let $F$ be a family of $k$-subsets of $\left[n\right]$ with matching number $s$. In the present paper, we determine the maximum $t$-diversity of $F$ for $1\le t\le 2s-1$ and $n\ge n_0(k,s)$.