On the Order Dimension of Convex Geometries

摘要

We study the order dimension of the lattice of closed sets for a convex geometry. We show that the lattice of closed subsets of the planar point set of Erdos and Szekeres from 1961, which is a set of 2~(n-2) points and contains no vertex set of a convex n-gon, has order dimension n － 1 and any larger set of points has order dimension at least n.