On the Order Dimension of Convex Geometries

TL;DR

This paper investigates the order dimension of convex geometries, revealing that certain large planar point sets have surprisingly low order dimension, with specific bounds related to Erdős and Szekeres point sets.

Contribution

It establishes bounds on the order dimension of convex geometries and constructs large examples with low order dimension, connecting geometric configurations to order-theoretic properties.

Findings

Erdős and Szekeres point sets have order dimension n-1

Large convex geometries can have very low order dimension

Order dimension increases with larger point sets

Abstract

We study the order dimension of the lattice of closed sets for a convex geometry. Further, we prove the existence of large convex geometries realized by planar point sets that have very low order dimension. We show that the planar point set of Erdos and Szekeres from 1961 which is a set of 2^(n-2) points and contains no convex n-gon has order dimension n - 1 and any larger set of points has order dimension strictly larger than n - 1.