# Convex independent sets and 7-holes in restricted planar point sets

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DOI: 10.1007/BF02187831

- Cite this article as:
- Valtr, P. Discrete Comput Geom (1992) 7: 135. doi:10.1007/BF02187831

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## Abstract

For a finite set*A* of points in the plane, let*q*(*A*) denote the ratio of the maximum distance of any pair of points of*A* to the minimum distance of any pair of points of*A*. For*k*>0 let*c*_{α}(*k*) denote the largest integer*c* such that any set*A* of*k* points in general position in the plane, satisfying\(q(A)< \alpha \sqrt k \) for fixed\(\alpha \geqslant \sqrt {2\sqrt 3 /\pi } \doteq 1.05\), contains at least*c* convex independent points. We determine the exact asymptotic behavior of*c*_{α}(*k*), proving that there are two positive constants*β*=*β*(α),*γ* such that*βk*^{1/3}≤*c*_{α}(*k*)≤*γk*^{1/3}. To establish the upper bound of*c*_{α}(*k*) we construct a set, which also solves (affirmatively) the problem of Alon*et al.* [1] about the existence of a set*A* of*k* points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points from*A*), satisfying\(q(A)< \alpha \sqrt k \). The construction uses “Horton sets,” which generalize sets without 7-holes constructed by Horton and which have some interesting properties.