Intersecting families with covering number three

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$ for all $F,F^{\mathrm{\prime }}\in \mathcal{F}$. In the present paper we prove that for $k\ge 7$, $n\ge 2k$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0px}{}{n-k}{k-1}\right)-\left(\genfrac{}{}{0px}{}{n-k-1}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n-2k}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n-k-2}{k-3}\right)+3$, the best possible upper bound which was proved in \cite{F80} subject to exponential constraints $n>n_0(k)$.