A superlinear lower bound on the number of 5-holes

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 h₅(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 h₅(n) have been of order Ω(n) and O(n²), respectively. We show that h₅(n)=Ω(nlog^4/5⁡n), obtaining the first superlinear lower bound on h₅(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 ℓ into two subsets, each of size at least 5 and not in convex position, then ℓ intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.