Abstract
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 $n)$ and~$O(n^2)$, respectively. We show that $h_5(n) = n4/5n)$, 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 $ into two subsets, each of size at least 5 and not in convex position, then $ intersects the convex hull of some 5-hole in~$P$. The proof of this result is computer-assisted.
Original language | English |
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Title of host publication | Proceedings of the 33rd European Workshop on Computational Geometry (EuroCG '17) |
Place of Publication | Malmö, Sweden |
Pages | 69-73 |
Number of pages | 5 |
Publication status | Published - 2017 |
Event | 33rd European Workshop on Computational Geometry: EuroCG 2017 - Malmö, Sweden Duration: 5 Apr 2017 → 7 Apr 2017 |
Conference
Conference | 33rd European Workshop on Computational Geometry |
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Abbreviated title | EuroCG 2017 |
Country/Territory | Sweden |
City | Malmö |
Period | 5/04/17 → 7/04/17 |