  In this manuscript we introduce and study an extended version of the minimal
dispersion of point sets, which has recently attracted considerable attention.
Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and
$k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the
largest box amidst a point set containing at most $k$ points. The minimal
$k$-dispersion is then given by the infimum over all possible point sets of
cardinality $n$. We provide both upper and lower bounds for the minimal
$k$-dispersion that coincide with the known bounds for the classical minimal
dispersion for a surprisingly large range of $k$'s.
