Many order types on integer grids of polynomial size

Two sets of labeled points $\{p_1,\dots ,p_n\}$ and $\{q_1,\dots ,q_n\}$ are of the same labeled order type if, for every $i,j,k$, the triples $(p_i,p_j,p_k)$ and $(q_i,q_j,q_k)$ have the same orientation. In the 1980's, Goodman, Pollack, and Sturmfels showed that (i) the number of labeled order types on n points is of order $n^{4n+o\left(n\right)}$, (ii) all order types can be realized with double-exponential integer coordinates, and that (iii) certain order types indeed require double-exponential integer coordinates. In 2018, Caraballo, Díaz-Báñez, Fabila-Monroy, Hidalgo-Toscano, Leaños, and Montejano showed that at least $n^{3n+o\left(n\right)}$ labeled n-point order types can be realized on an integer grid of polynomial size. In this article, we improve their result by showing that at least $n^{4n+o\left(n\right)}$ labeled n-point order types can be realized on an integer grid of polynomial size, which is asymptotically tight in the exponent. Finally we conclude that there are $n^{3n+o\left(n\right)}$ order types in the unlabeled setting and that $n^{3n+o\left(n\right)}$ of them can be realized on an integer grid of polynomial size.