Uniform intersecting families with large covering number

Abstract

A family $\mathcal{F}$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $\mathcal{F}$ is equal to $\tau$. Let $M(n,k,\tau )$ stand for the size of the largest intersecting family $\mathcal{F}$ of $k$-element subsets of $\{1,\dots ,n\}$ with covering number $\tau$. It is a classical result of Erd\H os and Lovász that $M(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\tau )$ for $n<k^2$ and large $\tau$. The results are quite surprising: For example, we show that $M(n,k,\tau )=(1-o(1))\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$, if $n=\lfloor k^{3/2}\rfloor$, and $\tau \le k-k^{3/4+o(1)}$ as $k\to \infty$; $M(n,k,\tau )<e^{-c{k}^{1/2}}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, if $n=\lfloor k^{3/2}\rfloor$ and $\tau >k-\frac{1}{2}{k}^{1/2}$.