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 $\tau>k-\frac 12k^{ 1/2}$.
Categories: math.CO cs.DM