  We show that any finite $S \subset \mathbb{R}^d$ in general position has
arbitrarily large supersets $T \supseteq S$ in general position with the
property that $T$ contains no empty convex polygon, or hole, with $C_d$ points,
where $C_d$ is an integer that depends only on the dimension $d$. This
generalises results of Horton and Valtr which treat the case $S = \emptyset$.
The key step in our proof, which may be of independent interest, is to show
that there are arbitrarily small perturbations of the set of lattice points
$[n]^d$ with no large holes.
