  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.
