Uniform intersecting families with large covering number

A family $mathcal F$ has covering number $tau$ if the size of the smallest set intersecting all sets from $mathcal F$ is equal to $s$. 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 ErdH os and Lovasz 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 $m(k^{3/2},k,tau) = (1-o(1)){n-1choose k-1}$ for $taule k-k^{3/4+o(1)}$. At the same time, $m(k^{3/2},k,tau)<e^{-ck^{1/2}}{nchoose k}$ if $tau>k-frac 12k^{1/2}$.