Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

For every integer r ≥ 3, we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly r minimal elements and exactly two maximal elements. The lattice has (r+2)² elements and the sublattice has (r+1)(r+2)/2 + 3 elements. At r = 3 the construction is the 25-element example of Bhattacharjee. Thus the number of minimal elements in such a complement is unbounded even when the convex dimension is fixed at three, in contrast to the common-minimum theorem in convex dimension two.