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 $\{1,\ldots,n\}$ with covering number $\tau$. It is a classical result of Erd\H os and Lov\'asz 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 $M(n,k,\tau) =(1-o(1)){n-1\choose k-1}$, if $n = \lfloor k^{3/2}\rfloor$, and $\tau\le k-k^{3/4+o(1)}$ as $k\to\infty$; $M(n,k,\tau) <e^{-ck^{1/2}}{n\choose k}$, if $n = \lfloor k^{3/2}\rfloor$ and $\tau>k-\frac 12k^{1/2}$.

翻译：设族$\mathcal F$的覆盖数为$\tau$，若与$\mathcal F$中所有集合相交的最小集合的大小等于$\tau$。令$M(n,k,\tau)$表示$\{1,\ldots,n\}$上$k$元子集构成的具有覆盖数$\tau$的最大相交族$\mathcal F$的大小。Erdős和Lovász的一个经典结果表明，对于任意$n$，有$M(n,k,k)\le k^k$。本文简要探讨了当$n<k^2$且$\tau$较大时$M(n,k,\tau)$的行为。结果颇为意外：例如，我们证明当$n = \lfloor k^{3/2}\rfloor$且$\tau\le k-k^{3/4+o(1)}$时（其中$k\to\infty$），有$M(n,k,\tau) =(1-o(1)){n-1\choose k-1}$；而当$n = \lfloor k^{3/2}\rfloor$且$\tau>k-\frac 12k^{1/2}$时，有$M(n,k,\tau) <e^{-ck^{1/2}}{n\choose k}$。