Improved bounds on the maximum diversity of intersecting families

Abstract

A family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ is called an intersecting family if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{F}$. If $\cap \mathcal{F}\ne \varnothing$ 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 $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ has diversity at most $\left(\genfrac{}{}{0ex}{}{n-3}{k-2}\right)$, 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.

arXiv admin note: substantial text overlap with arXiv:2207.05487