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arXiv · 1506.05505

Drawing the Horton Set in an Integer Grid of Minimum Size

Luis Barba·Frank Duque·Ruy Fabila-Monroy·Carlos Hidalgo-Toscano
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Abstract

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant.

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Luis Barba, Frank Duque, Ruy Fabila-Monroy, Carlos Hidalgo-Toscano. 2015-06-17. Drawing the Horton Set in an Integer Grid of Minimum Size. https://doi.org/10.1016/j.comgeo.2017.02.002

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