Drawing the Horton set in an integer grid of minimum size

In 1978 Erdős 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 12n12log⁡(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⋅n124log⁡(n/2), where c is a positive constant.