Chromatic numbers of copoint graphs of convex geometries

We study the copoint graph of a convex geometry. We give a family of copoint graphs for which the ratio of the chromatic number to the clique number can be arbitrarily large. For any natural numbers 1 < d < k , we study the existence of a number K d ( k ) so that the chromatic number of the copoint graph of a convex geometry on a set of at least K d ( k ) elements, with every d -element subset closed, has chromatic number at least k . Our results are analogues of results of Erdős and Szekeres for convex geometries realizable by point sets in R d , where cliques in the copoint graph correspond to subsets of points in convex position.