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
21βn21βlog(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 cβ n241βlog(n/2), where c is a positive constant