Intersecting families with covering number five

A family $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ is called intersecting if any two members of it have non-empty intersection. The covering number of $F$ is defined as the minimum integer p such that there exists $T\subset \{1,2,\dots ,n\}$ satisfying $\left|T\right|=p$ and $T\cap F\ne \varnothing$ for all $F\in F$. Define $m(n,k,p)$ as the maximum size of an intersecting family $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ with covering number at least p. The value of $m(n,k,p)$ is only known for $p=1,2,3,4$. About thirty years ago, $m(n,k,5)$ was determined asymptotically by the first author, Ota and Tokushige. In the present paper, we determine $m(n,k,5)$ for $k\ge 69$ and $n\ge 5k^6$.