Classification of maximum hittings by large families

For integers $r$ and $n$, where $n$ is sufficiently large, and for every set $X\subseteq [n]$ we determine the maximal left-compressed intersecting families $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{r}\right)$ which achieve maximum hitting with $X$ (i.e. have the most members which intersect $X$). This answers a question of Barber, who extended previous results by Borg to characterise those sets $X$ for which maximum hitting is achieved by the star.