Two  point configurations \{p_1,\ldots,p_n\} and \{q_1,\ldots,q_n\} are of the same 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 order types on n points is of order 4^{n+o(n)}, (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, D\'iaz-B\'a{\~n}ez, Fabila-Monroy, Hidalgo-Toscano, Lea{\~n}os, Montejano showed that n ^{ 3n-o (n)} order types can be realized on an integer grid of polynomial size. In this article, we show that n ^{ 4n-o (n)} order types can be realized on an integer grid of polynomial size, which is essentially best-possible .
Two labeled point configurations \{p_1,\ldots,p_n\} and \{q_1,\ldots,q_n\} are of the same 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 order types on n points is of order 4^{n+o(n)}, (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, D\'iaz-B\'a{\~n}ez, Fabila-Monroy, Hidalgo-Toscano, Lea{\~n}os, Montejano showed that at least n ^{ 3n+o (n)} 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 (n)} order types can be realized on an integer grid of polynomial size, which is essentially best possible .
A
labeled
R
n
at least n
R
3n-o
3n+o
R
show that n
improve their result by showing that at least n
R
4n-o
4n+o
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best-possible
best possible