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• W2596764499 startingPage "105236" @default.
• W2596764499 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 $Omega(n)$ and $O(n^2)$, respectively. We show that $h_5(n) = Omega(nlog^{4/5}{n})$, 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." @default.
• W2596764499 created "2017-03-23" @default.
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• W2596764499 date "2020-07-01" @default.
• W2596764499 modified "2023-09-25" @default.
• W2596764499 title "A superlinear lower bound on the number of 5-holes" @default.
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• W2596764499 doi "https://doi.org/10.1016/j.jcta.2020.105236" @default.
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