A family $\mathcal{ F}\subset \binom{ [n]}{ k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{ F}$. If $\cap \mathcal{ F}\neq \emptyset$ then $\mathcal{ F}$ is called a star. The diversity of an intersecting family $\mathcal{ F}$ is defined as the minimum number of $k$-sets in $\mathcal{ F}$, whose deletion results in a star. In the present paper, we prove that for $n>36k$ any intersecting family $\mathcal{ F}\subset \binom{ [n]}{ k}$ has diversity at most $\binom{ n-3}{ k-2}$, which improves the previous best bound $n>72k$ due to the first author. This result is derived from some strong bounds concerning the maximum degree of large intersecting families. Some related results are established as well.
Categories: math.CO