, Volume 30, Issue 3, pp 837-845

On the Order Dimension of Convex Geometries

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Abstract

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 Erdős 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.