Drawing the Horton Set in an Integer Grid of Minimum Size

TL;DR

This paper demonstrates how to embed the Horton set with no 7-gon in a small integer grid and establishes lower bounds on coordinate sizes for integer realizations of Horton-like point sets.

Contribution

It provides the first explicit integer grid realization of the Horton set with minimal size and analyzes the size bounds for any integer realization of Horton-like point sets.

Findings

Horton set can be realized with integer coordinates of size at most (1/2) n^{(1/2) log(n/2)}

Any integer realization of Horton-like sets contains a point with coordinate size at least c· n^{(1/24) log(n/2)}

The results establish bounds on the coordinate sizes for Horton set realizations.

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.