  Let $P$ be a finite set of points in the plane in general position, that is,
no three points of $P$ are on a common line. We say that a set $H$ of five
points from $P$ is a $5$-hole in $P$ if $H$ is the vertex set of a convex
$5$-gon containing no other points of $P$. For a positive integer $n$, let
$h_5(n)$ be the minimum number of 5-holes among all sets of $n$ points in the
plane in general position.
  Despite many efforts in the last 30 years, the best known asymptotic lower
and upper bounds for $h_5(n)$ have been of order $\Omega(n)$ and $O(n^2)$,
respectively. We show that $h_5(n) = \Omega(n\log^{4/5}{n})$, obtaining the
first superlinear lower bound on $h_5(n)$.
  The following structural result, which might be of independent interest, is a
crucial step in the proof of this lower bound. If a finite set $P$ of points in
the plane in general position is partitioned by a line $\ell$ into two subsets,
each of size at least 5 and not in convex position, then $\ell$ intersects the
convex hull of some 5-hole in $P$. The proof of this result is
computer-assisted.
