Many Order Types on Integer Grids of Polynomial Size

TL;DR

This paper improves the lower bound on the number of order types realizable on polynomial-sized integer grids, showing that nearly all order types can be realized with such coordinates, which is close to optimal.

Contribution

The authors significantly increase the known lower bound for realizable order types on polynomial grids, nearly matching the total number of order types.

Findings

At least $n^{4n+o(n)}$ order types can be realized on polynomial-sized integer grids.

This lower bound is essentially optimal, matching the total number of order types.

The result advances understanding of geometric realizability within polynomial coordinate bounds.

Abstract

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.