Point Sets with Small Integer Coordinates and with Small Convex Polygons

TL;DR

This paper demonstrates how to realize Erdős and Szekeres' convex polygon construction within a small integer grid, improving understanding of geometric configurations with integer coordinates.

Contribution

It provides a method to embed Erdős and Szekeres' point sets with small convex polygons into a polynomially bounded integer grid.

Findings

Construction fits within an O(n^2 log^3 n) grid

Sets have small integer coordinates

Convex polygons have at most log n + 1 vertices

Abstract

In 1935, Erd\H{o}s and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a set of $n:=2^{t-2}$ points in general position in the plane, such that every convex polygon with vertices in this set has at most $\log_2(n)+1$ vertices. In this paper we show how to realize their construction in an integer grid of size $O(n^2 \log_2(n)^3)$.