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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 fivepoints 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 theplane in general position.  Despite many efforts in the last 30 years, the best known asymptotic lowerand 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 thefirst superlinear lower bound on $h_5(n)$.  The following structural result, which might be of independent interest, is acrucial step in the proof of this lower bound. If a finite set $P$ of points inthe 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 theconvex hull of some 5-hole in $P$. The proof of this result iscomputer-assisted.

A superlinear lower bound on the number of 5-holes