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- W2072869041 abstract "In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n -gon with at most k points of X in its interior. Recall that h(n ,0 ) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2 (n+6)/4 -n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it." @default.
- W2072869041 created "2016-06-24" @default.
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- W2072869041 date "2003-08-01" @default.
- W2072869041 modified "2023-09-23" @default.
- W2072869041 title "Almost empty polygons" @default.
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- W2072869041 doi "https://doi.org/10.1556/sscmath.40.2003.3.1" @default.
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